Post on 15-Jan-2023
The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education
This work is licensed under the Creative Commons Attribution CC BY 4.0
MODULE 2
Logarithmic Functions
SECONDARY
MATH THREE
An Integrated Approach
Standard Teacher Notes
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
MODULE 2 - TABLE OF CONTENTS
LOGARITHMIC FUNCTIONS
2.1 Log Logic – A Develop Understanding Task Evaluate and compare logarithmic expressions. (F.BF.5, F.LE.4) Ready, Set, Go Homework: Logarithmic Functions 2.1
2.2 Falling Off a Log – A Solidify Understanding Task Graph logarithmic functions with transformations (F.BF.3, F.BF.5, F.IF.7e) Ready, Set, Go Homework: Logarithmic Functions 2.2
2.3 Chopping Logs – A Solidify Understanding Task Explore properties of logarithms. (F.IF.8, F.LE.4) Ready, Set, Go Homework: Logarithmic Functions 2.3
2.4 Log-Arithm-etic – A Practice Understanding Task Use log properties to evaluate expressions (F.IF.8, F.LE.4) Ready, Set, Go Homework: Logarithmic Functions 2.4
2.5 Powerful Tens – A Practice Understanding Task Solve exponential and logarithmic functions in base 10 using technology. (F.LE.4) Ready, Set, Go Homework: Logarithmic Functions 2.5
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.1 Log Logic A Develop Understanding Task
WebeganthinkingaboutlogarithmsasinversefunctionsforexponentialsinTrackingtheTortoise.Logarithmicfunctionsareinterestingandusefulontheirown.Inthenextfewtasks,wewillbeworkingonunderstandinglogarithmicexpressions,logarithmicfunctions,andlogarithmicoperationsonequations.
Weshowedtheinverserelationshipbetweenexponentialandlogarithmicfunctionsusingadiagramliketheonebelow:
Wecouldsummarizethisrelationshipbysaying:
2" = 8so,log)8 = 3
Logarithmscanbedefinedforanybaseusedforanexponentialfunction.Base10ispopular.Usingbase10,youcanwritestatementslikethese:
10- = 10 so, log-.10 = 1
10) = 100 so, log-.100 = 2
10" = 1000 so, log-.1000 = 3
CC B
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Input Output
/(1) = 23 /4-(1) = log)1
1 = 3 3 2" = 8
1
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Thenotationmayseedifferent,butyoucanseetheinversepatternwheretheinputsandoutputsswitch.
Thenextfewproblemswillgiveyouanopportunitytopracticethinkingaboutthispatternandpossiblymakeafewconjecturesaboutotherpatternsrelatedtologarithms.
Placethefollowingexpressionsonthenumberline.Usethespacebelowthenumberlinetoexplainhowyouknewwheretoplaceeachexpression.
1. A.log"3 B.log"9C.log" -" D.log"1 E.log" -6
Explain:____________________________________________________________________________________________________
2. A.log"81 B.log-.100 C.log78D.log825 E.log)32
Explain:____________________________________________________________________________________________________
3. A.log:7 B.log69 C.log--1D.log-.1
Explain:____________________________________________________________________________________________________
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SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
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4. A.log) <-=> B.log-. <-
-...>C.log8 <--)8>D.log? <
-?>
Explain:____________________________________________________________________________________________________
5. A.log=16 B.log)16C.log716D.log-?16
Explain:____________________________________________________________________________________________________
6. A.log)5 B.log810 C.log?1 D.log85 E.log-.5
Explain:____________________________________________________________________________________________________
7. A.log-.50 B. log-.150 C. log-.1000 D. log-.500
Explain:____________________________________________________________________________________________________
8. A.log"3) B.log854) C.log?6. D.log=44- E.log)2"
Explain:____________________________________________________________________________________________________
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SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
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Basedonyourworkwithlogarithmicexpressions,determinewhethereachofthesestatementsisalwaystrue,sometimestrue,ornevertrue.Ifthestatementissometimestrue,describetheconditionsthatmakeittrue.Explainyouranswers.
9. ThevalueoflogB 1ispositive.
Explain:____________________________________________________________________________________________________
10. logB 1isnotavalidexpressionifxisanegativenumber.
Explain:____________________________________________________________________________________________________
11. logB 1 = 0foranybase,b>0.
Explain:____________________________________________________________________________________________________
12. logB C = 1foranyb>0.
Explain:____________________________________________________________________________________________________
13. log) 1<log" 1foranyvalueofx.
Explain:____________________________________________________________________________________________________
14. logDCE = Fforanyb>0.
Explain:____________________________________________________________________________________________________
4
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.1 Log Logic – Teacher Notes A Develop Understanding Task
Purpose:
Thepurposeofthistaskistodevelopstudents’understandingoflogarithmicexpressionsandto
makesenseofthenotation.Inadditiontoevaluatinglogexpressions,studentwillcompare
expressionsthattheycannotevaluateexplicitly.Theywillalsousepatternstheyhaveseeninthe
taskandthedefinitionofalogarithmtojustifysomepropertiesoflogarithms.
CoreStandardsFocus:
F.BF.5(+)Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethis
relationshiptosolveproblemsinvolvinglogarithmsandexponents.
F.LE.4Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddare
numbersandthebasebis2,10,ore;evaluatethelogarithmusingtechnology.
NoteforF.LE.4:Considerextendingthisunittoincludetherelationshipbetweenpropertiesof
logarithmsandpropertiesofexponents,suchastheconnectionbetweenthepropertiesofexponents
andthebasiclogarithmpropertythatlogxy=logx+logy.
RelatedStandards:F.BF.4
StandardsforMathematicalPractice:
SMP2–Reasonabstractlyandquantitatively
SMP8–Lookforandexpressregularityinrepeatedreasoning
Vocabulary:baseofalogarithm,argumentofalogarithm
TheTeachingCycle:
Launch(WholeClass):
Beginbyworkingthrougheachoftheexamplesonpage1ofthetaskwithstudents.Tellthemthat
sinceweknowthatlogarithmicfunctionsandexponentialfunctionsareinverses,thedefinitionofa
logarithmis:
IfC3 = FthenlogB F = 1forb>0
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
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Keepthisrelationshippostedwherestudentscanrefertoitduringtheirworkonthetask.
Explore(SmallGroup):
Thetaskbeginswithexpressionsthatwillgenerateintegervalues.Inthebeginning,encourage
studentstousethepatternexpressedinthedefinitiontohelpfindthevalues.Iftheydon’tknow
thepowersofthebasenumbers,theymayneedtousecalculatorstoidentifythem.Forinstance,if
theyareaskedtoevaluatelog)32,theymayneedtousethecalculatortofind2: = 32.(Author’snote:Ihopethiswon’tbethecase,buttheemphasisinthistaskisonreasoning,notonarithmetic
skill.)Thinkingaboutthesevalueswillhelptoreviewintegerexponents.
Startingat#5,thereareexpressionsthatcanonlybeestimatedandplacedonthenumberlineina
reasonablelocation.Don’tgivestudentsawaytousethecalculatortoevaluatetheseexpressions
directly;againtheemphasisisonreasoningandcomparing.
Asyoumonitorstudentsastheywork,keeptrackofstudentsthathaveinterestingjustificationsfor
theiranswersonproblems#9–15sothattheycanbeincludedintheclassdiscussion.
Discuss(WholeClass):
Beginthediscussionwith#2.Foreachlogexpression,writetheequivalentexponentialequation
likeso:
log"81 = 43> = 81
Thiswillgivestudentspracticeinseeingtherelationshipbetweenexponentialfunctionsand
logarithmicfunctions.Placeeachofthevaluesonthenumberline.
Movethediscussionto#4andproceedinthesameway,givingstudentsabrush-uponnegative
exponents.
Next,workwithquestion#5.Sincestudentscan’tcalculatealloftheseexpressionsdirectly,they
willhavetouselogictoordertheexpressions.Onestrategyistofirstputtheexpressionsinorder
fromsmallesttobiggestbasedontheideathatthebiggerthebase,thesmallertheexponentwill
needtobetoget16.(Besurethisideaisgeneralizedbytheendofthediscussionof#5.)Oncethe
numbersareinorder,thentheapproximatevaluescanbeconsideredbaseduponknownvaluesfor
aparticularbase.
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
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Workon#7next.Inthisproblem,thebasesarethesame,buttheargumentsaredifferent.The
expressionscanbeorderedbasedontheideathatforagivenbase,b>1,thegreatertheargument,
thegreatertheexponentwillneedtobe.
Finally,workthrougheachofproblems9–15.Thisisanopportunitytodevelopanumberofthe
propertiesoflogarithmsfromthedefinitions.Afterstudentshavejustifiedeachoftheproperties
thatarealwaystrue(#10,11,12,and14),theseshouldbepostedintheclassroomasagreed-upon
propertiesthatcanbeusedinfuturework.
AlignedReady,Set,Go:LogarithmicFunctions2.1
SECONDARY MATH III // MODULE 2
LOGRITHMIC FUNCTIONS –
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2.1
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REA DY Topic:**Graphing*exponential*equations**Graph$each$function$over$the$domain$ −! ≤ ! ≤ ! .$*1.**! = 2!!!!*
*2.**! = 2 ∙ 2! *
*
3.**! = !!!*
*
4.**! = 2 !!!*
*
* * * ***
5.***Compare*graph*#1*to*graph*#2.$Multiplying*by*2*should*generate*a*dilation*of*the*graph,*but*the*graph*looks*like*it*has*been*translated*vertically.**How*do*you*explain*that?*
*
*
6.***Compare*graph*#3*to*graph*#4.$$Is*your*explanation*in*#5*still*valid*for*these*two*graphs?*Explain.*
SET Topic:**Writing*the*logarithmic*form*of*an*exponential*equation.*
Definition$of$Logarithm:$$$$For$all$positive$numbers$a,$where$a$≠$1,$and$all$positive$numbers$x,$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! = !"#!!$means$the$same$as$x"="!!.$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$(Note$the$base$of$the$exponent$and$the$base$of$the$logarithm$are$both$a.)$
$
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READY, SET, GO! ******Name* ******Period***********************Date*
5
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LOGRITHMIC FUNCTIONS –
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2.1
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7.***Why*is*it*important*that*the*definition*of*logarithm*states*that*the*base*of*the*logarithm*does*not*equal*1?**
*8.***Why*is*it*important*that*the*definition*states*that*the*base*of*the*logarithm*is*positive?*
*
9.***Why*is*it*necessary*that*the*definition*states*that*x*in*the*expression*!"#!!**is*positive?**
Write$the$following$exponential$equations$in$logarithmic$form.$
Exponential$form$ Logarithmic$form$ Exponential$form$ Logarithmic$form$
*10.**5! = 625**
* *11.**3! = 9*
*
12.**!!!!= 8* * 13.**4!! = !
!"**
*14.***10! = 10000*
* *15.**!!*=*x*
*
*
16.**Compare*the*exponential*form*of*an*equation*to*the*logarithmic*form*of*an*equation.**What*part*of*the*exponential*equation*is*the*answer*to*the*logarithmic*equation?*
Topic:**Considering*values*of*logarithmic*functions**Answer$the$following$questions.*If$yes,$give$an$example$or$the$answer.$$If$no,$explain$why$not.$
17.**Is*it*possible*for*a*logarithm*to*equal*a*negative*number?**
18.**Is*it*possible*for*a*logarithm*to*equal*zero?*
19.**Does*!"#!0*have*an*answer?*
20.**Does*!"#!1*have*an*answer?*
21.**Does*!!"!!!*have*an*answer?$
6
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LOGRITHMIC FUNCTIONS –
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2.1
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GO Topic:**Reviewing*properties*of*Exponents*
Write$each$expression$as$an$integer$or$a$simple$fraction.*
22.***270* 23.***11(Y6)0* 24.***−3!!*
*
25.****4!!* 26.*****!!!!* * 27.*****
!!!!*
*
28.****3 !"!!!!
!* 29.*****
!!!!* * 30.*****
!"!!!!! *
*
7
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.2 Falling Off a Log A Solidify Understanding Task
1. Constructatableofvaluesandagraphforeachofthefollowingfunctions.Besuretoselectatleasttwovaluesintheinterval0<x<1.
a) !(#) = log)#
b) *(#) = log+#
CC B
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8
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
c) ℎ(#) = log-#
d) .(#) = log/0#
2. Howdidyoudecidewhatvaluestouseforxinyourtable?
3. Howdidyouusethexvaluestofindtheyvaluesinthetable?
9
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
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4. Whatsimilaritiesdoyouseeinthegraphs?
5. Whatdifferencesdoyouobserveinthegraphs?
6. Whatistheeffectofchangingthebaseonthegraphofalogarithmicfunction?
Let’sfocusnowon.(#) = log/0#sothatwecanusetechnologytoobservetheeffectsofchangingparametersonthefunction.Becausebase10isaverycommonlyusedbaseforexponentialandlogarithmicfunctions,itisoftenabbreviatedandwrittenwithoutthebase,likethis:1(2) = 3452.
7. Usetechnologytograph7 = log #.Howdoesthegraphcomparetothegraphthatyouconstructed?
8. Howdoyoupredictthatthegraphof7 = 8 + log #willbedifferentfromthegraphof7 = log #?
9. Testyourpredictionbygraphing7 = 8 + log #forvariousvaluesofa.Whatistheeffectofaonthegraph?Makeageneralargumentforwhythiswouldbetrueforalllogarithmicfunctions.
10. Howdoyoupredictthatthegraphof7 = log(# + :)willbedifferentfromthegraphof7 = log #?
10
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
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11. Testyourpredictionbygraphing7 = log(# + :)forvariousvaluesofb.• Whatistheeffectofaddingb?
• Whatwillbetheeffectofsubtractingb(oraddinganegativenumber)?
• Makeageneralargumentforwhythisistrueforalllogarithmicfunctions.
12. Writeanequationforeachofthefollowingfunctionsthataretransformationsof!(#) = log) #.
a.
b.
11
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
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13. Graphandlabeleachofthefollowingfunctions:a. !(#) = 2 +log)(# − 1)
b. *(#) = −1 + log)(# + 2)
14. Comparethetransformationofthegraphsoflogarithmicfunctionswiththetransformation
ofthegraphsofquadraticfunctions.
12
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
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2. 2 Falling Off a Log – Teacher Notes A Solidify Understanding Task
Purpose:Thepurposeofthistaskistobuildonstudents’understandingofalogarithmicfunction
astheinverseofanexponentialfunctionandtheirpreviousworkindeterminingvaluesfor
logarithmicexpressionstofindthegraphsoflogarithmicfunctionsofvariousbases.Studentsuse
technologytoexploretransformationswithloggraphsinbase10andthengeneralizethe
transformationstootherbases.
CoreStandardsFocus:
F.IF.7.aGraphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,and
trigonometricfunctions,showingperiod,midline,andamplitude.
F.BF.5 (+)Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethis
relationshiptosolveproblemsinvolvinglogarithmsandexponents.
NoteforF.BF:Usetransformationsoffunctionstofindmoreoptimummodelsasstudentsconsider
increasinglymorecomplexsituations.
ForF.BF.3,notetheeffectofmultipletransformationsonasinglefunctionandthecommoneffectof
eachtransformationacrossfunctiontypes.Includefunctionsdefinedonlybyagraph.
ExtendF.BF.4atosimplerational,simpleradical,andsimpleexponentialfunctions;connectF.BF.4a
toF.LE.4.
RelatedStandards:F.LE.4
StandardsforMathematicalPractice:
SMP2–Reasonabstractlyandquantitatively
SMP5–Useappropriatetoolsstrategically
Vocabulary:verticalasymptote
NotetoTeachers:Accesstographingtechnologyisnecessaryforthistask.
ADesmosactivityforthisistaskisavailableat:https://tinyurl.com/MVPMath3Lesson2-2. It
canbefoundatDesmosbysearchingfor:MVPMathIII,2.2FallingOffaLog.Theactivity
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
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takesstudentsthroughquestions7-11,allowingthemtousetechnologytoexplorethe
verticalandhorizontalshiftsonthegraphsofalogarithmicfunction.
TheTeachingCycle
Launch(WholeClass):
Beginclassbyremindingstudentsoftheworktheydidwithlogexpressionsintheprevioustask
andsolicitingafewexponentialandlogstatementslikethis:
5/ = 125BC logD 125 = 3
Encouragetheuseofdifferentbasestoremindstudentsthatthesamedefinitionworksforall
bases,b>1.Tellstudentsthatinthistasktheywillusewhattheyknowaboutinversestohelp
themcreatetablesandgraphlogfunctions.
Explore(SmallGroup):
Monitorstudentsastheyworktoensuretheyarecompletingbothtablesandgraphsforeach
function.Somestudentsmaychoosetographtheexponentialfunctionandthenreflectitoverthe
; = $linetogetthegraphbeforecompletingthetable.Watchforthisstrategyandbepreparedtohighlightitduringthediscussion.Findingpointsonthegraphfor0 < $ < 1mayprovedifficultforstudentssincenegativeexponentsareoftendifficult.Ifstudentsarestuck,remindthemthatitmay
beeasiertofindpointsontheexponentialfunctionandthenswitchthemfortheloggraphs.
Discuss(WholeClass):
Beginthediscussionwiththegraphof&($) = log-$.Askastudentthatusedtheexponentialfunction; = 2F andswitchedthexandyvaluestopresenttheirgraph.Thenhaveastudentthatstartedbycreatingatabledescribehowtheyobtainedthevaluesinthetable.Asktheclassto
identifyhowthetwostrategiesareconnected.Bynow,studentsshouldbeabletoarticulatethe
ideathatpowersof2areeasyvaluestothinkaboutandthevalueofthelogexpressionwillbethe
exponentineachcase.
Movethediscussiontoquestion#4,thesimilaritiesbetweenthegraphs.Studentswillprobably
speakgenerallyabouttheshapesbeingalike.Inthediscussionsofsimilarities,besurethatthe
moretechnicalfeaturesofthegraphsemerge:
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
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• Thepoint(1,0)isincluded
• Thedomainis(0,∞)
• Therangeis(-∞,∞)
• Thefunctionisincreasingovertheentiredomain.
Askstudentstoconnecteachofthesefeatureswiththedefinitionofalogarithmandpropertiesof
inversefunctions.Atthispoint,introducetheideaofaverticalasymptote.Weknowthat
logarithmicfunctionsarenotdefinedfor$ = 0becauseexponentialfunctionsapproach,butneveractuallyreach0.Sincelogarithmicfunctionsareinversesofexponentialfunctions,wewouldexpect
theirgraphstobereflectionsoverthe; = $line.Alsohelpstudentstounderstandwhythey-valuesofalogarithmicfunctionbecomeverylargenegativenumbersas$-valuesbecomecloserto0.
Askwhatconclusionstheycoulddrawabouttheeffectofchangingthebaseongraph.Howdo
theseconclusionsconnecttothestrategiestheyusedtoorderlogexpressionswithdifferentbases
intheprevioustask?
Askstudentshowthegraphsweretransformedwhenanumberisaddedoutsidethelogfunctions
versusinsidetheargumentofthelogfunction.Studentsshouldnoticethatthisisjustlikeother
functionsthattheyarefamiliarwithsuchasquadraticfunctions.
AlignedReady,Set,Go:LogarithmicFunctions2.2
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LOGRITHMIC FUNCTIONS –
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2.2
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REA DY Topic:**Solving*simple*logarithmic*equations**Find$the$answer$to$each$logarithmic$equation.$$Then$explain$how$each$equation$supports$the$statement,$“The%answer%to%a%logarithmic%equation%is%always%the%exponent.”%$
1. !"#!625 =%
*
2. !"#!243 =*
*
3. !"#!0.2 =*
*
4. !"#!81 =*
*
5. !"#1,000,000 =*
*
6. !"#!!! =*
*
SET Topic:**Exploring*transformations*on*logarithmic*functions**Answer$the$questions$about$each$graph.**7.***
*a. What*is*the*value*of*x*when*! ! = 0?*b. What*is*the*value*of*x*when*! ! = 1?*c. What*is*the*value*of*! ! *when*! = 2?*d. What*will*be*the*value*of*x*when*! ! = 4?*e. What*is*the*equation*of*this*graph?*
*8.***
*a.*****What*is*the*value*of*x*when*! ! = 0?***************b.*****What*is*the*value*of*x*when*! ! = 1?*******************c.******What*is*the*value*of*! ! *when*! = 9?***************d.******What*will*be*the*value*of*x*when*! ! = 4?*****e.******What*is*the*equation*of*this*graph?*
***
READY, SET, GO! ******Name* ******Period***********************Date*
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2.2
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9.**Use*the*graph*and*the*table*of*
values*for*the*graph*to*write*the*
equation*of*the*graph.*
Explain*which*numbers*in*the*
table*helped*you*the*most*to*
write*the*equation.*
*
*
*
10.**Use*the*graph*and*the*table*of*
values*for*the*graph*to*write*the*
equation*of*the*graph.*
Explain*which*numbers*in*the*
table*helped*you*the*most*to*
write*the*equation.*
*
*
* GO Topic:**Using*the*power*to*a*power*rule*with*exponents**Simplify$each$expression.$$Answers$should$have$only$positive$exponents.$$11.*** 2! !***** * 12.*** !! !*** * 13.*** !! !!***** * 14.*** 2!! !*****
*
*
*
15.*** !!! !********* 16.***** !!! !!******** 17.****!! ∙ !! !***** 18.***!!! ∙ !! !*
*
*
*
19.*** !! !! ∙ !!"* 20.*** 5!! !***** * 21.*** 6!! !***** * 22.*** 2!!!! !*********
14
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.3 Chopping Logs A Solidify Understanding Task
AbeandMarywereworkingontheirmathhomeworktogetherwhenAbehasabrilliantidea!
Abe:IwasjustlookingatthislogfunctionthatwegraphedinFallingOffALog:
! = log'() + +).
IstartedtothinkthatmaybeIcouldjust“distribute”thelogsothatIget:
! = log' ) + log' +.
IguessI’msayingthatIthinktheseareequivalentexpressions,soIcouldwriteitthisway:
log'() + +) = log' ) + log' +
Mary:Idon’tknowaboutthat.LogsaretrickyandIdon’tthinkthatyou’rereallydoingthesamethinghereaswhenyoudistributeanumber.
1. Whatdoyouthink?HowcanyouverifyifAbe’sideaworks?
2. IfAbe’sideaworks,givesomeexamplesthatillustratewhyitworks.IfAbe’sideadoesn’twork,giveacounter-example.
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SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Abe:Ijustknowthatthereissomethinggoingonwiththeselogs.Ijustgraphed-()) = log'(4)).Hereitis:
It’sweirdbecauseIthinkthatthisgraphisjustatranslationof! = log' ).Isitpossiblethattheequationofthisgraphcouldbewrittenmorethanoneway?
3. HowwouldyouanswerAbe’squestion?Arethereconditionsthatcouldallowthesamegraphtohavedifferentequations?
Mary:Whenyousay,“atranslationof! = log' )”doyoumeanthatitisjustaverticalorhorizontalshift?Whatcouldthatequationbe?
4. Findanequationfor-())thatshowsittobeahorizontalorverticalshiftof! = log' ).
16
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Mary:Iwonderwhytheverticalshiftturnedouttobeup2whenthexwasmultipliedby4.Iwonderifithassomethingtodowiththepowerthatthebaseisraisedto,sincethisisalogfunction.Let’strytoseewhathappenswith! = log'(8))and! = log'(16)).
5. Trytowriteanequivalentequationforeachofthesegraphsthatisaverticalshiftof! = log' ).
a) ! = log'(8)) Equivalentequation:____________________________________________
b.! = log'(16)) Equivalentequation:____________________________________________
17
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Mary:Ohmygosh!IthinkIknowwhatishappeninghere!Here’swhatweseefromthegraphs:log'(4)) = 2 + log' )
log'(8)) = 3 + log' )
log'(16)) = 4 + log' )
Here’sthebrilliantpart:Weknowthatlog' 4 = 2, log' 8 = 3, and log' 16 = 4.So:
log'(4)) = log' 4 + log' )
log'(8)) = log' 8 + log' )
log'(16)) = log' 16 + log' )
Ithinkitlookslikethe“distributive”thingthatyouweretryingtodo,butsinceyoucan’treallydistributeafunction,it’sreallyjustalogmultiplicationrule.Iguessmyrulewouldbe:
log'(9+) = log' 9 + log' +
6. HowcanyouexpressMary’sruleinwords?
7. Isthisstatementtrue?Ifitis,givesomeexamplesthatillustratewhyitworks.Ifitisnottrueprovideacounterexample.
18
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Mary:So,Iwonderifasimilarthinghappensifyouhavedivisioninsidetheargumentofalogfunction.I’mgoingtotrysomeexamples.Ifmytheoryworks,thenallofthesegraphswilljustbeverticalshiftsof! = log' ).
8. HereareAbe’sexamplesandtheirgraphs.TestAbe’stheorybytryingtowriteanequivalentequationforeachofthesegraphsthatisaverticalshiftof! = log' ).
a) ! = log' :;
<= Equivalentequation:_____________________________________________
b) ! = log' :;
>= Equivalentequation:__________________________________________
9. UsetheseexamplestowritearulefordivisioninsidetheargumentofalogthatisliketherulethatMarywroteformultiplicationinsidealog.
19
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
10. Isthisstatementtrue?Ifitis,givesomeexamplesthatillustratewhyitworks.Ifitisnottrueprovideacounterexample.
Abe:You’redefinitelybrilliantforthinkingofthatmultiplicationrule.ButI’mageniusbecauseI’veusedyourmultiplicationruletocomeupwithapowerrule.Let’ssaythatyoustartwith:
log'( )?)
Reallythat’sthesameashaving: log'() ∙ ) ∙ ))
So,Icoulduseyourmultiplyingruleandwrite: log' ) + log' ) + log' )Inoticethatthereare3termsthatareallthesame.Thatmakesit: 3 log' )
Somyruleis: log'()?) = 3 log' )
Ifyourruleistrue,thenIhaveprovenmypowerrule.
Mary:Idon’tthinkit’sreallyapowerruleunlessitworksforanypower.Youonlyshowedhowitmightworkfor3.
Abe:Oh,goodgrief!Ok,I’mgoingtosaythatitcanbeanynumberx,raisedtoanypower,k.Mypowerruleis:
log'()A) = B log' )
Areyousatisfied?
11. ProvideanargumentaboutAbe’spowerrule.Isittrueornot?
20
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Abe:BeforewewintheNobelPrizeformathematicsIsupposethatweneedtothinkaboutwhetherornottheserulesworkforanybase.
12. Thethreerules,writtenforanybaseb>1are:LogofaProductRule: CDEF(GH) = CDEF G + CDEF HLogofaQuotientRule: CDEF :
G
H= = CDEF G − CDEF H
LogofaPowerRule: CDEF(GJ) = J CDEF G
Makeanargumentforwhytheseruleswillworkinanybaseb>1iftheyworkforbase2.
13. Howaretheserulessimilartotherulesforexponents?Whymightexponentsandlogshavesimilarrules?
21
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.3 Chopping Logs – Teacher Notes
A Solidify Understanding Task NotetoTeachers:Accesstographingtechnologyisnecessaryforthistask.
Purpose:Thepurposeofthistaskistousestudentunderstandingofloggraphsandlogexpressionstoderivepropertiesoflogarithms.Inthetaskstudentsareaskedtofindequivalent
equationsforgraphsandthentogeneralizethepatternstoestablishtheproduct,quotient,and
powerrulesforlogarithms.CoreStandardsFocus:
F.IF.8.Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealand
explaindifferentpropertiesofthefunction.
F.LE.4.Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddare
numbersandthebasebis2,10,ore;evaluatethelogarithmusingtechnology.
NotetoF.LE.4:Considerextendingthisunittoincludetherelationshipbetweenpropertiesof
logarithmsandpropertiesofexponents,suchastheconnectionbetweenthepropertiesofexponents
andthebasiclogarithmpropertythatlog(xy)=logx+logy.
RelatedStandards:F.BF.5
StandardsforMathematicalPractice:
SMP5–Useappropriatetoolsstrategically
SMP7–Lookforandmakeuseofstructure
TheTeachingCycle:
Note:Thenatureofthistasksuggestsamoreguidedapproachfromtheteacherthanmanytasks.Themostproductiveclassroomconfigurationmightbepairs,sothatstudentscaneasilyshifttheir
attentionbackandforthfromwholegroupdiscussiontotheirownwork.
Launch(WholeClass):Beginthetaskbyintroducingtheequation:log'() + +) = log' ) + log' +.Askstudentswhythismightmakesense.Expecttohearthattheyhave“distributed”thelog.
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Withoutjudgingthemeritsofthisidea,askstudentshowtheycouldtesttheclaim.Whentheideatotestparticularnumberscomesup,setstudentstoworkonquestions#1and#2.Afterstudents
havehadachancetoworkon#2,askastudentthathasanexampleshowingthestatementtobe
untruetosharehis/herwork.Youmayneedtohelprewritethestudent’sworksothatthestatementsareclearbecausethisisastrategythatstudentswillwanttousethroughoutthetask.
Explore(SmallGroup):Askstudentstoturntheirattentiontoquestion#3.Basedontheirworkwithloggraphsinprevioustasks,theyhavenotseenthegraphofalogfunctionwithamultiplierin
theargument,like-()) = log'(4)).Askstudentshowthe4mightaffectthegraph.BecauseAbethinksthefunctionisaverticalshiftof! = log' ),askstudentwhattheequationwouldlooklikewithaverticalshiftsothattheygeneratetheideathattheverticalshifttypicallylookslike
! = 9 + log' ).Askstudentstoinvestigatequestions#3and#4.Asstudentsareworking,lookforastudentthathasredrawnthex-axisoruseastraightedgetoshowthetranslationofthegraph.
Discuss(WholeClass):Whenstudentshavehadenoughtotimetofindtheverticalshift,havea
studentdemonstratehowtheywereabletotellthatthefunctionisaverticalshiftup2.Itwillhelp
movetheclassforwardinthenextpartofthetaskifastudentdemonstratesredrawingthex-axissoitiseasytoseethatthegraphisatranslation,butnotadilationof! = log' ).Afterthisdiscussion,havestudentsworkonquestions#5,6,and7.
Explore(SmallGroup):Whilestudentsareworking,listenforstudentswhoareabletodescribe
thepatternMaryhasnoticedinthetask.EncouragestudentstotestMary’sconjecturewithsomenumbers,justastheydidinthebeginningofthetask.
Discuss(WholeClass):AskseveralstudentstostateMary’sruleintheirownwords.Trytocombinethestudentstatementsintosomethinglike:“Thelogofaproductisthesumofthelogs”or
givethemthisstatementandaskthemtodiscusshowitdescribesthepatterntheyhavenoticed.Haveseveralstudentsshowexamplesthatprovideevidencethatthestatementistrue,butremind
studentsthatafewexamplesdon’tcountasaproof.Afterthisdiscussion,askstudentstocomplete
therestofthetask.
Explore(SmallGroup):Supportstudentsastheyworktorecognizethepatternsandexpressarulein#9asbothanequationandinwords.Studentsmayhavedifficultywiththenotation,soask
themtostatetheruleinwordsfirst,andthenhelpthemtowriteitsymbolically.
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Discuss(WholeClass):Theremainingdiscussionshouldfolloweachofthequestionsinthetaskfrom#9onward.Asthediscussionprogresses,showstudentexamplesofeachrule,bothto
provideevidencethattheruleistrueandalsotopracticeusingtherule.Emphasizereasoningthat
helpsstudentstoseethatthelogrulesareliketheexponentrulesbecauseoftherelationshipbetweenlogsandexponents.
AlignedReady,Set,Go:LogarithmicFunctions2.3
SECONDARY MATH III // MODULE 2
LOGRITHMIC FUNCTIONS –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.3
Need$help?$$Visit$www.rsgsupport.org$
REA DY Topic:**Recalling*fractional*exponents**Write$the$following$with$an$exponent.$$Simplify$when$possible.$$1.*** !! *
*
2.*** !!! * 3.*** !!! * 4.*** 8!!! *
5.*** 125!!!
*
6.*** 8! !! * 7.*** 9!!! * 8.*** 75!!*
$Rewrite$with$a$fractional$exponent.$$Then$find$the$answer.$*9.***!"#! 3! = **
*
*10.***!"#! 4! = *
*11.***!"#! 343! = *
*12.***!"#! 3125! = *
SET Topic:**Using*the*properties*of*logarithms*to*expand*logarithmic*expressions**Use$the$properties$of$logarithms$to$expand$the$expression$as$a$sum$or$difference,$and/or$constant$multiple$of$logarithms.$$(Assume*all*variables*are*positive.)***13.***!"#!7!*
*14.***!"#!10!*
*15.***!"#! !!*
*16.***!"#! !!*****
17.***!"#!!! * 18.***!"#!9!!* 19.***!"#! 7! !* 20.***!"#! !******
READY, SET, GO! ******Name* ******Period***********************Date*
22
SECONDARY MATH III // MODULE 2
LOGRITHMIC FUNCTIONS –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.3
Need$help?$$Visit$www.rsgsupport.org$
*21.***!"#! !"#! * 22.***!"#! ! !
!! * 23.***!"#! !!!!!! * 24.***!"#! !!
!!!! *******
GO Topic:**Writing*expressions*in*exponential*form*and*logarithmic*form**Convert$to$logarithmic$form.$$
25.***2! = 512* * * 26.***10!! = 0.01*** * 27.*** !!!!= !
!*********Write$in$exponential$form.$
*28.***!"#!2 = ! !!**** * 29.***!"#!
!3 = −1***** * 30.***!"#!
!
!!"# = 3*
23
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.4 Log-Arithm-etic A Practice Understanding Task
AbeandMaryarefeelinggoodabouttheirlogrulesandbraggingabouttheirmathematicalprowesstoalloftheirfriendswhenthisexchangeoccurs:
Stephen:Iguessyouthinkyou’reprettysmartbecauseyoufiguredoutsomelogrules,butIwanttoknowwhatthey’regoodfor.
Abe:Well,we’veseenalotoftimeswhenequivalentexpressionsarehandy.Sometimeswhenyouwriteanexpressionwithavariableinitinadifferentwayitmeanssomethingdifferent.
1. Whataresomeexamplesfromyourpreviousexperiencewhereequivalentexpressionswereuseful?
Mary:IwasthinkingabouttheLogLogictaskwhereweweretryingtoestimateandordercertainlogvalues.Iwaswonderingifwecoulduseourlogrulestotakevaluesweknowandusethemtofindvaluesthatwedon’tknow.
Forinstance:Let’ssayyouwanttocalculatelog$ 6.Ifyouknowwhatlog$ 2andlog$ 3arethenyoucanusetheproductruleandsay:
log$(2 ∙ 3) = log$ 2 + log$ 3
Stephen:That’sgreat.Everyoneknowsthatlog$ 2 = 1,butwhatislog$ 3?
Abe:Oh,Isawthissomewhere.Uh,log$ 3 =1.585.SoMary’sideareallyworks.Yousay:
log$(2 ∙ 3) = log$ 2 + log$ 3
= 1 + 1.585
= 2.585
log$ 6=2.585
2. Basedonwhatyouknowaboutlogarithms,explainwhy2.585isareasonablevalueforlog$ 6.
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SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Stephen:Oh,oh!I’vegotone.Icanfigureoutlog$ 5likethis:
log$(2 + 3) = log$ 2 + log$ 3
= 1 + 1.585
= 2.585
log$ 5=2.585
3. CanStephenandMarybothbecorrect?Explainwho’sright,who’swrong(ifanyone)andwhy.
Nowyoucantryapplyingthelogrulesyourself.Usethevaluesthataregivenandtheonesthatyouknowbydefinition,likelog$ 2 = 1,tofigureouteachofthefollowingvalues.Explainwhatyoudidinthespacebeloweachquestion.
log$ 3 =1.585 log$ 5 =2.322 log$ 7 =2.807
Thethreerules,writtenforanybaseb>1are:
LogofaProductRule: 4567(89) = 4567 8 + 4567 9LogofaQuotientRule: 4567 :
89; = 4567 8 − 4567 9
LogofaPowerRule: 4567(8=) = = 4567 8
4. log$ 9 = ________________________________________________
5. log$ 10 = ________________________________________________
25
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6. log$ 12 = ________________________________________________
7. log$ :@A;= ________________________________________________
8. log$ :AB@; = ________________________________________________
9. log$ 486 = ________________________________________________
10. Giventheworkthatyouhavejustdone,whatothervalueswouldyouneedtofigureoutthevalueofthebase2logforanynumber?
26
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Sometimesthinkingaboutequivalentexpressionswithlogarithmscangettricky.Considereachofthefollowingexpressionsanddecideiftheyarealwaystrueforthenumbersinthedomainofthelogarithmicfunction,sometimestrue,ornevertrue.Explainyouranswers.Ifyouanswer“sometimestrue”,describetheconditionsthatmustbeinplacetomakethestatementtrue.
11. logD 5 − logD E = logD :FG; _______________________________________________________
12. log 3 − log E − log E = log : AGH;_____________________________________________________
13. log E − log 5 = IJKGIJK F
_____________________________________________________
14. 5 log E = log EF _____________________________________________________
15. 2 log E + log 5 = log(E$ + 5) _____________________________________________________
16. L$log E = log √E _____________________________________________________
17. log(E − 5) = IJK GIJK F
_____________________________________________________
27
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.4 Log-Arithm-etic – Teacher Notes A Practice Understanding Task
Purpose:
Thepurposeofthistaskistoextendstudentunderstandingoflogpropertiesandusingthe
propertiestowriteequivalentexpressions.Inthebeginningofthetask,studentsaregivenvalues
ofafewlogexpressionsandaskedtouselogpropertiesandknownvaluesoflogexpressionstofind
unknownvalues.Thisisanopportunitytoseehowtheknownlogvaluescanbeusedandto
practiceusinglogarithmsandsubstitution.Inthesecondpartofthetask,studentsareaskedto
determineifthegivenequationsarealwaystrue(inthedomainoftheexpression),sometimestrue,
ornevertrue.Thisgivesstudentsanopportunitytoworkthroughsomecommonmisconceptions
aboutlogpropertiesandtowriteequivalentexpressionsusinglogs.
CoreStandardsFocus:
F.IF.8.Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealand
explaindifferentpropertiesofthefunction.
F.LE.4.Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddare
numbersandthebasebis2,10,ore;evaluatethelogarithmusingtechnology.
NotetoF.LE.4:Considerextendingthisunittoincludetherelationshipbetweenpropertiesof
logarithmsandpropertiesofexponents,suchastheconnectionbetweenthepropertiesofexponents
andthebasiclogarithmpropertythatlog(xy)=logx+logy.
StandardsforMathematicalPractice:
SMP2–Reasonabstractlyandquantitatively
SMP6–Attendtoprecision
TheTeachingCycle
Launch(WholeClass):Launchthetaskbyreadingthroughthescenarioandaskingstudentsto
workproblem#1.Followitbyadiscussionoftheiranswers,pointingoutthatequivalentforms
oftenhavedifferentmeaningsinastorycontextandtheycanbehelpfulinsolvingequationsand
graphing.Followthisshortdiscussionbyhavingstudentsworkproblems#2and#3,then
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
discussingthemasaclass.Thepurposeofquestion#2istodemonstratehowtousethelogrules
tofindvalues,andemphasizehowtheycanusethedefinitionofalogarithmtodetermineifthe
valuetheyfindisreasonable.Afterdiscussingthesetwoproblems,studentsshouldbereadytouse
thepropertiestofindvaluesoflogs.Havestudentsworkquestions#4-10beforecomingbackfora
discussion.
Explore(SmallGroup):Asstudentsareworking,theymayneedsupportinfindingcombinations
offactorstousesotheycanapplythelogproperties.Youmaywanttoremindthemofusingfactor
treesorasimilarstrategyforbreakingdownanumberintoitsfactors.Watchfortwostudentsthat
useadifferentcombinationoffactorstofindthevaluetheyarelookingfor.Asyouaremonitoring
studentwork,besuretheyareusinggoodnotationtocommunicatehowtheyarefindingthe
values.
Discuss(WholeClass):Discussafewoftheproblems,selectingthosethatcausedcontroversy
amongstudentsastheyworked.Foreachproblem,besuretodemonstratethewaytousenotation
andthelogpropertiestofindthevalues.Anexamplemightbe:
log$ :AB@ ; = log$ 30 − log$ 7
= log$(5 ∙ 2 ∙ 3) − log$ 7
= log$ 5 + log$ 2 + log$ 3 − log$ 7
=2.322+1+1.585–2.807
=2.1
Afterfindingeachvalue,discusswhetherornottheanswerisreasonable.Afterafewofthese
problems,turnstudents’attentiontotheremainderofthetask.
Explore(SmallGroup):Supportstudentsastheyworkinmakingsenseofthestatementsand
verifyingthem.Thestatementsaredesignedtobringoutmisconceptions,sodiscussionamong
studentsshouldbeencouraged.Thereareseveralpossiblestrategiesforverifyingtheseequations,
includingusingthelogpropertiestomanipulateonesideoftheequationtomatchtheotheror
tryingtoputinnumberstothestatement.Lookforbothtypesofstrategiessothatthenumerical
approachcanprovideevidence,butthealgebraicapproachcanprove(ordisprove)thestatement.
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Discuss(WholeGroup):Again,selectproblemsfordiscussionthathavegeneratedcontroversyor
exposedmisconceptions.Itwilloftenbeusefultotestthestatementwithnumbers,althoughthat
maybedifficultforstudentsinsomecases.Encouragestudentstocitethelogpropertytheyare
usingastheymanipulatethestatementstoshowequivalence.Forstatementsthatarenevertrue,
askstudentshowtheymightcorrectthestatementtomakeittrue.
11.Alwaystrue 12.Alwaystrue 13.Nevertrue
14.Alwaystrue 15.Nevertrue 16.Alwaystrue
17.Nevertrue
AlignedReady,Set,Go:LogarithmicFunctions2.4
SECONDARY MATH III // MODULE 2
LOGRITHMIC FUNCTIONS –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.4
Needhelp?Visitwww.rsgsupport.org
READY Topic:SolvingsimpleexponentialandlogarithmicequationsYouhavesolvedexponentialequationsbeforebasedontheideathat!" = !$, &'!)*+),$&'" = $.
Youcanusethesamelogiconlogarithmicequations.,+.!" = ,+.!$, &'!)*+),$&'" = $
Rewriteeachequationsothatyousetupaone-to-onecorrespondencebetweenalloftheparts.Thensolveforx.Example:Originalequation
a.)30 = 81
b.)34567 − 34565 = 0
Rewrittenequation:
30 = 3;
34567 = 34565
Solution:
7 = 4
7 = 5
1.30=; = 243
2. ?6
0= 8
3. @;
0=
6A
B;
4.34567 − 345613 = 0
5.3456 27 − 4 − 34568 = 0 6.3456 7 + 2 − 345697 = 0
7.EFG 60EFG ?;
= 1 8.EFG H0I?EFG 6J
= 1 9.EFG HKLM
EFG B6H= 1
READY, SET, GO! Name PeriodDate
28
SECONDARY MATH III // MODULE 2
LOGRITHMIC FUNCTIONS –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.4
Needhelp?Visitwww.rsgsupport.org
SET Topic:RewritinglogsintermsofknownlogsUsethegivenvaluesandthepropertiesoflogarithmstofindtheindicatedlogarithm.
Donotuseacalculatortoevaluatethelogarithms.
Given:NOP QR ≈ Q. TNOP U ≈ V. WNOP X ≈ V. Y
10.Findlog H]
11.Findlog 25
12.Findlog ?6
13.Findlog 80 14.Findlog ?
B;
Given,+.^T ≈ V. R,+.^U ≈ Q. U
15.Find345@16
16.Find345@108
17.Find345@@
H`
18.Find345@]
?H 19.Find345@486
20.Find345@18 21.Find345@120 22.Find345@@6
;H
29
SECONDARY MATH III // MODULE 2
LOGRITHMIC FUNCTIONS –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.4
Needhelp?Visitwww.rsgsupport.org
GO Topic:Usingthedefinitionoflogarithmtosolveforx.Useyourcalculatorandthedefinitionoflogx(recallthebaseis10)tofindthevalueofx.(Roundyouranswersto4decimals.)
23.logx=-3 24.logx = 1 25.logx = 0
26.logx = ?6 27.logx = 1.75 28.logx = −2.2
29.logx = 3.67 30.logx = @;31.logx = 6
30
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.5 Powerful Tens A Practice Understanding Task
TablePuzzles1. Usethetablestofindthemissingvaluesofx:
c.Whatequationscouldbewritten,intermsofxonly,foreachoftherowsthataremissingthexinthetwotablesabove?
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x ! = #$%-2
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1 10 50 1003 1000
b.
x ! = ((#$%) 0.30 3 94.872 300 1503.56
d.
x ! = ,-.%0.01 -2
-1
10 1
1.6
100 2
e.
x ! = ,-.(% + () -2
-2.9 -1
0.3
7 1
3
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SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
f.Whatequationscouldbewritten,intermsofxonly,foreachoftherowsthataremissingthexinthetwotablesabove?
2. Whatstrategydidyouusetofindthesolutionstoequationsgeneratedbythetablesthatcontainedexponentialfunctions?
3. Whatstrategydidyouusetofindthesolutionstoequationsgeneratedbythetablesthatcontainedlogarithmicfunctions?
GraphPuzzles4.Thegraphofy=1012 isgivenbelow.Usethegraphtosolvetheequationsforxandlabelthesolutions.
a. 40 = 1012 4 = _____LabelthesolutionwithanAonthegraph.
b. 1012 = 104 = _____LabelthesolutionwithaBonthegraph.
c. 1012 = 0.14 = _____LabelthesolutionwithaConthegraph.
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LOGARITHMIC FUNCTIONS – 2.5
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5.Thegraphofy= − 2 + log 4isgivenbelow.Usethegraphtosolvetheequationsforxandlabelthesolutions.
a. −2 + log 4 = −24 = _____
LabelthesolutionwithanAonthegraph.
b. −2 + log 4 = 04 = _____
LabelthesolutionwithaBonthegraph.
c.−4 = −2 + log 4 4 = _____LabelthesolutionwithaConthegraph.d.−1.3 = −2 + log 4 4 = _____LabelthesolutionwithaDonthegraph.e.1 = −2 + log 4 4 = _____
6.Arethesolutionsthatyoufoundin#5exactorapproximate?Why?
EquationPuzzles:Solveeachequationforx:
7. 102=10,000 8. 125 = 102 9. 102>? = 347
10. 5(102>?) = 0.25 11. 10121A = ABC 12. −(102>?) = 16
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2.5 Powerful Tens – Teacher Notes A Practice Understanding Task
Note:Calculatorsorothertechnologywithbase10logarithmicandexponentialfunctionsare
requiredforthistask.
Purpose:
Thepurposeofthistaskistodevelopstudentideasaboutsolvingexponentialequationsthat
requiretheuseoflogarithmsandsolvinglogarithmicequations.Thetaskbeginswithstudents
findingunknownvaluesintablesandwritingcorrespondingequations.Inthesecondpartofthe
task,studentsusegraphstofindequationsolutions.Finally,studentsbuildontheirthinkingwith
tablesandgraphstosolveequationsalgebraically.Allofthelogarithmicandexponentialequations
areinbase10sothatstudentscanusetechnologytofindvalues.
CoreStandardsFocus:
F.LE.4.Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddare
numbersandthebasebis2,10,ore;evaluatethelogarithmusingtechnology.
StandardsforMathematicalPractice:
SMP5–Useappropriatetoolsstrategically
SMP6–Attendtoprecision
TheTeachingCycle:
Launch(WholeClass):
Remindstudentsthattheyareveryfamiliarwithconstructingtablesforvariousfunctions.In
previoustasks,theyhaveselectedvaluesforxandcalculatedthevalueofybaseduponanequation
orotherrepresentation.Theyhavealsoconstructedgraphsbaseduponhavinganequationoraset
ofxandyvalues.Inthistasktheywillbeusingtablesandgraphstoworkinreverse,findingthex
valueforagiveny.
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
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Explore(SmallGroup):
Monitorstudentsastheyworkandlistentotheirstrategiesforfindingthemissingvaluesofx.As
theyareworkingonthetablepuzzles,encouragethemtoconsiderwritingequationsasawayto
tracktheirstrategies.Inthegraphpuzzles,theywillfindtheycanonlygetapproximateanswerson
afewequations.Encouragethemtousethegraphtoestimateavalueandtointerpretthesolution
intheequation.Thepurposeofthetablesandgraphsistohelpstudentsdrawupontheirthinking
fromprevioustaskstosolvetheequations.Remindstudentstoconnecttheideasastheyworkon
theequationpuzzles.
Discuss(WholeClass):
Startthediscussionwithastudentthathaswrittenandsolvedanequationforthethirdrowin
tableb.Theequationwrittenshouldbe:
FG.HI = )($%&)
Askthestudenttodescribehowtheywrotetheequationandthentheirstrategyforsolvingit.Be
suretohavestudentsdescribetheirthinkingabouthowtounwindthefunctionasthestepsare
trackedontheequation.Asktheclasswherethispointwouldbeonagraphofthefunction.Ask
studentswhatthegraphofthefunctionwouldlooklike,andtheyshouldbeabletodescribeabase
10exponentialfunctionwithaverticalstretchof3.
Movethediscussiontotable“e”,focusingonthelastrowofthetable.Again,havestudentswrite
theequation:) = -./(& + ))
Askthepresentingstudenttodescribehis/herthinkingabouthowtofindthevalueof!inthetableand,onceagain,trackthestepsalgebraically.Thereareacoupleoflikelymistakesmadeby
studentswhohavetriedtosolvethisequationalgebraically.Iftheyariseduringyourobservation
ofstudents,discussthemhere.Again,connectthesolutiontothegraphofthefunction.Students
shouldbenoticingthatsincelogsandexponentialsareinversefunctions,exponentialequationscan
besolvedwithlogsandlogequationsaresolvedwithexponentials.
Movethediscussiontothegraphof7 = 1023 .Askstudentstodescribehowtheyusedthegraphtofindthesolutionto“a”.Askstudentshowtheycouldcheckthesolutionintheequation.Doesthe
SECONDARY MATH III // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
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solutiontheyfoundwiththegraphmakesense?Howwouldtheysolvethisequationwithouta
graph?Trackthestepsalgebraically,showingsomethinglikethefollowing:
40 = 1023
log 40 = log(10−!) 1.602 = −!
(Makesurestudentscanexplainthisstep,bothusingthecalculatorandsimplifyingtherightsideof
theequation.Itwouldbeusefulifstudentsnoticedtheycouldusethelogpropertiestorewritethe
rightsideoftheequationas−!(log 10)inadditiontousingthedefinitionofthelogarithm.)
! = −1.602
Finally,askstudentstoshowsolutionstoasmanyoftheequationpuzzlesthattimewillallow.In
everycase,besurestudentscandescribehowtheyuselogstoundotheexponentialandthattheir
notationmatchestheirthinking.
AlignedReady,Set,Go:LogarithmicFunctions2.5
SECONDARY MATH III // MODULE 2
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2.5
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READY Topic:ComparingthegraphsoftheexponentialandlogarithmicfunctionsThegraphsof! " = 10&()*+ " = log "areshowninthesamecoordinateplane.
Makealistofthecharacteristicsofeachfunction.
1. ! " = 10&
2.+ " = log "
Eachquestionbelowreferstothegraphsofthefunctions/ 0 = 1203456 0 = 789 0.State
whethertheyaretrueorfalse.Iftheyarefalse,correctthestatementsothatitistrue.
__________ 3.Everygraphoftheform+ " = log "willcontainthepoint(1,0).
__________ 4.Bothgraphshaveverticalasymptotes.
__________ 5.Thegraphsof! " ()*+ " havethesamerateofchange.
__________ 6.Thefunctionsareinversesofeachother.
__________ 7.Therangeof! " isthedomainof+ " .
__________ 8.Thegraphof+ " willneverreach3.
READY, SET, GO! Name PeriodDate
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2.5
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SET Topic:Solvinglogarithmicequations(base10)bytakingthelogofeachsideEvaluatethefollowinglogarithms9.log 10 10.log 10;< 11.log 10<= 12.log 10&
13.>?+@3= 14.>?+B8;@ 15.>?+DD11@< 16.>?+EFG
Youcanusethispropertyoflogarithmstohelpyousolvelogarithmicequations.
*Note:Thispropertyonlyworkswhenthebaseofthelogarithmmatchesthebaseoftheexponent.
SolvetheequationsbyinsertingHI64onbothsidesoftheequation.(Youwillneedacalculator.)
17.10G = 4.305 18.10G = 0.316 19.10G = 14,52120.10G = 483.059
GO Topic:SolvingequationsinvolvingcompoundinterestFormulaforcompoundinterest:IfPdollarsisdepositedinanaccountpayinganannualrateof
interestrcompounded(paid)ntimesperyear,theaccountwillcontainP = Q 1 + S4
4Tdollars
aftertyears.21.Howmuchmoneywilltherebeinanaccountattheendof10yearsif$3000isdepositedat6%annualinterestcompoundedasfollows: (Assumenowithdrawalsaremade.) a.) annually b.) semiannually c.) quarterly d.) daily(Usen=365.)22.Findtheamountofmoneyinanaccountafter12yearsif$5,000isdepositedat7.5%annualinterestcompoundedasfollows: (Assumenowithdrawalsaremade.) a.) annually b.) semiannually c.) quarterly d.) daily(Usen=365.)
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