Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 1

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 2

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 3

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 4

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 5

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 6

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 7

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 8

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 2

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 3

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 4

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 5

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 6

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 7

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 8

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 3

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 4

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 5

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 6

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 7

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 8

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 4

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 5

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 6

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 7

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 8

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 5

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 6

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 7

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 8

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 6

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 7

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 8

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 7

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 8

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 8

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 9

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 10

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 11

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3

Sketch and analyze the graph of each function Describe its domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

1 f (x) = ndashex + 7

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing

D = (ndash ) R = (ndash 0) y-intercept ndashe7 asymptote x-axis decreasing for (ndash

)

x minus7 minus5 minus4 minus3 y minus1 minus74 minus201 minus546

2

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

List the domain range intercepts asymptotes end behavior and where the function is increasing or decreasing D = (ndash ) R = (ndash4 ) y-intercept ndash2 x-intercept 136 asymptote y = ndash4

increasing for (ndash )

x minus4 minus1 0 2 3 4 y minus374 minus28 minus2 156 526 1143

Use the graph of f (x) to describe the transformation that results in the graph of g (x) Then sketch the graphs of f (x) and g (x)

3

SOLUTION

This function is of the form Therefore the graph of g(x) is the graph of f (x) translated 3 units to the

right and 4 units up The translation to the right is indicated by the subtraction of 3 in the exponent The translation upis indicated by the addition of 4 Use the graphs of the functions to confirm this transformation

4 f (x) = 5x

SOLUTION

This function is of the form f (x) = 5x Therefore the graph of g(x) is the graph of f (x) reflected in the x-axis

reflected in the y-axis and translated 2 units down The reflection in the x-axis is indicated by the negative coefficient in front of the 5 The reflection in the y-axis is indicated by the negative exponent The translation down is indicated by the subtraction of 2 Use the graphs of the functions to confirm this transformation

5 MULTIPLE CHOICE For which function is

A f (x) = ndash2 3minusx

B

C f (x) = ndashlog8 (x ndash 5)

D f (x) = log3 (ndashx) ndash 6

SOLUTION

For option A as x approaches infinity 3minusx approaches 0 so the function approaches 0 For option B as x

approaches infinity approaches 0 so the function approaches 0 For option D as x approaches infinity log3

(minusx) is undefined For option C as x approaches infinity log8 (x minus 5) approaches infinity and minus log8 (x minus 5)

approaches negative infinity The correct choice is C

Evaluate each expression

6 log3

SOLUTION

7 log32 2

SOLUTION

8 log 1012

SOLUTION

9

SOLUTION

Sketch the graph of each function10 f (x) = ndashlog4 (x + 3)

SOLUTION Convert the base of the equation

Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 1 2 y undef 0 minus05 minus079 minus1 minus116

11 g(x) = log (ndashx) + 5

SOLUTION Evaluate the function for several x-values in its domain

Then use a smooth curve to connect each of these ordered pairs

x minus3 minus2 minus1 0 y 548 530 5 undef

12 FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8 for 12 years making no other deposits or withdrawals a What will be your account balance if the interest is compounded monthly b What will be your account balance if the interest is compounded continuously c If your investment is compounded daily about how long will it take for it to be worth double the initial amount

SOLUTION a

b

c

Expand each expression

13 log6 36xy2

SOLUTION

14 log3

SOLUTION

15 GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the

energy produced and E0 is a constant

a An earthquake with a magnitude of 71 hit San Francisco in 1989 Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake b In 1906 San Francisco had an earthquake registering 825 How many times as much energy did the 1906 earthquake produce as the 1989 earthquake

SOLUTION a If E is the energy produced by the 1989 earthquake then 10 times this energy is 10E Use the 1989 data to solve

for E0 then replace E with 10E to find the magnitude of an earthquake with 10 times the energy

Now solve for R

The magnitude is about 78 b For this problem we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E Let the 1906 earthquake = nE

The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake

Condense each expression16 2 log4 m + 6 log4 n ndash 3(log4 3 + log4 j )

SOLUTION

17 1 + ln 3 ndash 4 ln x

SOLUTION

Solve each equation

18 3x + 8

= 92x

SOLUTION

19 e2x ndash 3ex + 2 = 0

SOLUTION

x = ln 2 or 0

20 log x + log (x ndash 3) = 1

SOLUTION

The logarithm of a negative has no real solution Thus x = 5

21 log2 (x ndash 1) + 1 = log2 (x + 3)

SOLUTION

22 MULTIPLE CHOICE Which equation has no solution

F ex = e

ndashx

G 2x ndash 1

= 3x + 1

H log5 x = log9 x

J log2 (x + 1) = log2 x

SOLUTION For option F x can equal 0 Option G can be solved using the natural log For option H x can equal 1 For option J x+ 1 must equal x so it has no solution

For Exercises 23 and 24 complete each step a Find an exponential or logarithmic function to model the datab Find the value of each model at x = 20

23

SOLUTION a An exponential regression cannot be completed with negative y-values Calculate the logarithmic regression

The rounded logarithmic regression equation is f (x) = 820 ndash 511ln x b Using the full regression equation f (20) asymp minus711

24

SOLUTION a The exponential regression has the stronger correlation coefficient

The rounded exponential regression equation is f (x) = 29(11)x

Using the full exponential regression equation f (20) asymp 2122

25 CENSUS The table gives the US population between 1790 and 1940 Let 1780 = 0

a Linearize the data assuming a quadratic model Graph the data and write an equation for a line of best fit b Use the linear model to find a model for the original data Is a quadratic model a good representation of populationgrowth Explain

SOLUTION

a What linearizing data according to a quadratic model use (x )

Plot the data

Find the regression

The rounded regression equation is y = 007x

b Substitute for and solve for y

The rounded regression equation is y = 00041x2 + 01331x + 10816 Population growth generally does not grow

without bounds Typically it will start to level off and plateau For these reasons a logistic model would be more appropriate

eSolutions Manual - Powered by Cognero Page 12

Practice Test - Chapter 3