Post on 18-Mar-2023
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