Logarithmic Functions - Mathematics Vision Project · Ready, Set, Go Homework: Logarithmic...

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

ALGEBRA II

An Integrated Approach

ALGEBRA II // 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.5, F.IF.7a) 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


LOGARITHMIC FUNCTIONS – 2.1


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

Thenotationmayseedifferent,butyoucanseetheinversepatternwheretheinputsandoutputsswitch.

Thenextfewproblemswillgiveyouanopportunitytopracticethinkingaboutthispatternandpossiblymakeafewconjecturesaboutotherpatternsrelatedtologarithms.

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/(1) = 23 /4-(1) = log)1

1 = 3 3 2" = 8

1




Placethefollowingexpressionsonthenumberline.Usethespacebelowthenumberlinetoexplainhowyouknewwheretoplaceeachexpression.

1. A.log"3 B.log"9C.log" 6-"7 D.log"1 E.log" 6-87

Explain:____________________________________________________________________________________________________

2. A.log"81 B.log-.100 C.log98D.log:25 E.log)32

Explain:____________________________________________________________________________________________________

3. A.log<7 B.log89 C.log--1D.log-.1

Explain:____________________________________________________________________________________________________

4. A.log) 6->7 B.log-. 6-

-...7C.log: 6--):7D.log? 6

-?7

Name Date

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2TheMathWorksheetSite.com

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0 1 2 3 4 5

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0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5TheMathWorksheetSite.com

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0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5


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Explain:____________________________________________________________________________________________________

5. A.log>16 B.log)16C.log916D.log-?16

Explain:____________________________________________________________________________________________________

6. A.log)5 B.log:10 C.log?1 D.log:5 E.log-.5

Explain:____________________________________________________________________________________________________

7. A.log-.50 B. log-.150 C. log-.1000 D. log-.500

Explain:____________________________________________________________________________________________________

8. A.log"3) B.log:54) C.log?6. D.log>44- E.log)2"

Explain:____________________________________________________________________________________________________

Basedonyourworkwithlogarithmicexpressions,determinewhethereachofthesestatementsisalwaystrue,sometimestrue,ornevertrue.Ifthestatementissometimestrue,describetheconditionsthatmakeittrue.Explainyouranswers.

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0 1 2 3 4 5

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0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5


Name Date

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0 1 2 3 4 5

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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




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




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.




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



2.1


Needhelp?Visitwww.rsgsupport.org

READY Topic:GraphingexponentialequationsGrapheachfunctionoverthedomain{−# ≤ % ≤ #}.1.( = 2+

2.( = 2 ∙ 2+

3.( = -.

/0+

4.( = 2-.

/0+

5.Comparegraph#1tograph#2.Multiplyingby2shouldgenerateadilationofthegraph,butthegraphlookslikeithasbeentranslatedvertically.Howdoyouexplainthat?

6.Comparegraph#3tograph#4.Isyourexplanationin#5stillvalidforthesetwographs?Explain.

SET Topic:Writingthelogarithmicformofanexponentialequation.

DefinitionofLogarithm:Forallpositivenumbersa,wherea≠1,andallpositivenumbersx,

2 = 3456%meansthesameasx=62.

(Notethebaseoftheexponentandthebaseofthelogarithmarebotha.)

–5 5

20

18

16

14

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10

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–2–5 5

20

18

16

14

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10

8

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–2

READY, SET, GO! Name PeriodDate

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2.1



7.Whyisitimportantthatinthedefinitionoflogarithmitisstatedthatthebaseofthelogarithmdoesnotequal1?

8.Whyisitimportantthatthedefinitionofalogarithmstatesthatthebaseofthelogarithmis

positive?

9.Whyisitnecessarythatthedefinitionstatesthatxintheexpression3456%ispositive?

Writethefollowingexponentialequationsinlogarithmicform.

Exponentialform Logarithmicform Exponentialform Logarithmicform

10.58 = 625

11.3/ = 9

12.-.

/0<== 8 13.4</ =

.

.@

14.108 = 10000

15.CD =x

16.Comparetheexponentialformofanequationtothelogarithmicformofanequation.Whatpartoftheexponentialequationistheanswertothelogarithmicequation?

Topic:ConsideringvaluesoflogarithmicfunctionsAnswerthefollowingquestions.Ifyes,giveanexampleoftheanswer.Ifno,explainwhynot.

17.Isitpossibleforalogarithmtoequalanegativenumber?

18.Isitpossibleforalogarithmtoequalzero?

19.DoesEFG+0haveananswer?

20.DoesEFG+1haveananswer?

21.DoesEFG+HIhaveananswer?

6



2.1



GO Topic:ReviewingpropertiesofExponents

Writeeachexpressionasanintegerorasimplefraction.

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2.2 Falling Off a Log A Solidify Understanding Task

1. Constructatableofvaluesandagraphforeachofthefollowingfunctions.Besuretoselectatleasttwovaluesintheinterval0 < $ < 1.

a) &($) = log-$

b) .($) = log/$

c) ℎ($) = log1$

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d) 2($) = log34$

2. Howdidyoudecidewhatvaluestousefor$inthetable?

3. Howdidyouusethe$valuestofindtheyvaluesinthetable?

4. Whatsimilaritiesdoyouseeinthegraphs?

9




5. Whatdifferencesdoyouobserveinthegraphs?

6. Whatistheeffectofchangingthebaseonthegraphofalogarithmicfunction?

Let’sfocusnowon2($) = log34$sothatwecanusetechnologytoobservetheeffectsofchangingparametersonthefunction.Becausebase10isacommonlyusedbaseforexponentialandlogarithmicfunctions,itisoftenabbreviatedandwrittenwithoutthebase,likethis:5(6) =7896.

7. Usetechnologytograph; = log $.Howdoesthegraphcomparetothegraphthatyouconstructed?

8. Howdoyoupredictthatthegraphof; = < + log $willbedifferentfromthegraphof; = log $?

9. Testyourpredictionbygraphing; = < + log $forvariousvaluesofa.Whatistheeffectofaonthegraph?Makeageneralargumentforwhythiswouldbetrueforalllogarithmicfunctions.

10. Howdoyoupredictthatthegraphof; = log($ + >)willbedifferentfromthegraphof; = log $?

11. Testyourpredictionbygraphing; = log($ + >)forvariousvaluesofb.• Whatistheeffectofaddingb?

• Whatwillbetheeffectofsubtractingb(oraddinganegativenumber)?

10




• Makeageneralargumentforwhythisistrueforalllogarithmicfunctions.

12. Writeanequationforeachofthefollowingfunctionsthataretransformationsof&($) = log- $.

a)

b)

11




13. Graphandlabeleachofthefollowingfunctions:a) &($) = 2 +log-($ − 1)

b) .($) = −1 + log-($ + 2)

14. Comparethetransformationofthegraphsoflogarithmicfunctionswiththetransformation

ofthegraphsofquadraticfunctions.

12




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



SMP5–Useappropriatetoolsstrategically

Vocabulary:verticalasymptote

NotetoTeachers:Accesstographingtechnologyisnecessaryforthistask.




ADesmosactivityforthisistaskisavailableat:https://tinyurl.com/MVPMath3Lesson2-2. It

canbefoundatDesmosbysearchingfor:MVPMathIII,2.2FallingOffaLog.Theactivity

takesstudentsthroughquestions7-11,allowingthemtousetechnologytoexplorethe

verticalandhorizontalshiftsonthegraphsofalogarithmicfunction.

TheTeachingCycle

Launch(WholeClass):

Beginclassbyremindingstudentsoftheworktheydidwithlogexpressionsintheprevioustask

andsolicitingafewexponentialandlogstatementslikethis:

5/ = 125BC logD 125 = 3

Encouragetheuseofdifferentbasestoremindstudentsthatthesamedefinitionworksforall

bases,b>1.Tellstudentsthatinthistasktheywillusewhattheyknowaboutinversestohelp

themcreatetablesandgraphlogfunctions.


Monitorstudentsastheyworktoensuretheyarecompletingbothtablesandgraphsforeach

function.Somestudentsmaychoosetographtheexponentialfunctionandthenreflectitoverthe

; = $linetogetthegraphbeforecompletingthetable.Watchforthisstrategyandbepreparedtohighlightitduringthediscussion.Findingpointsonthegraphfor0 < $ < 1mayprovedifficultforstudentssincenegativeexponentsareoftendifficult.Ifstudentsarestuck,remindthemthatitmay

beeasiertofindpointsontheexponentialfunctionandthenswitchthemfortheloggraphs.


Beginthediscussionwiththegraphof&($) = log-$.Askastudentthatusedtheexponentialfunction; = 2F andswitchedthexandyvaluestopresenttheirgraph.Thenhaveastudentthatstartedbycreatingatabledescribehowtheyobtainedthevaluesinthetable.Asktheclassto

identifyhowthetwostrategiesareconnected.Bynow,studentsshouldbeabletoarticulatethe

ideathatpowersof2areeasyvaluestothinkaboutandthevalueofthelogexpressionwillbethe

exponentineachcase.




Movethediscussiontoquestion#4,thesimilaritiesbetweenthegraphs.Studentswillprobably

speakgenerallyabouttheshapesbeingalike.Inthediscussionsofsimilarities,besurethatthe

moretechnicalfeaturesofthegraphsemerge:

• 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.




2.2



READY Topic:SolvingsimplelogarithmicequationsFindtheanswertoeachlogarithmicequation.Thenexplainhoweachequationsupportsthestatement,“Theanswertoalogarithmicequationisalwaystheexponent.”

1. !"#$625 =

2. !"#)243 =

3. !"#$0.2 =

4. !"#.81 =

5. !"#1,000,000 =

6. !"#234 =

SET Topic:ExploringtransformationsonlogarithmicfunctionsAnswerthequestionsabouteachgraph.7.

a. Whatisthevalueofxwhen5(3) = 0?b. Whatisthevalueofxwhen5(3) = 1?c. Whatisthevalueof5(3)when3 = 2?d. Whatwillbethevalueofxwhen5(3) = 4?e. Whatistheequationofthisgraph?

8.

a.Whatisthevalueofxwhen5(3) = 0?b.Whatisthevalueofxwhen5(3) = 1?c.Whatisthevalueof5(3)when3 = 9?d.Whatwillbethevalueofxwhen5(3) = 4?e.Whatistheequationofthisgraph?


13



2.2



9.Usethegraphandthetableofvaluesforthegraphtowritetheequationofthegraph.Explainwhichnumbersinthetablehelpedyouthemosttowritetheequation.

10.Usethegraphandthetableofvaluesforthegraphtowritetheequationofthegraph.Explainwhichnumbersinthetablehelpedyouthemosttowritetheequation.

GO Topic:UsingthepowertoapowerrulewithexponentsSimplifyeachexpression.Answersshouldhaveonlypositiveexponents.11.(2)): 12.(3)); 13.(<))=; 14.(2)>):

15.(?=4)) 16.(@=))=; 17.3; ∙ (3$); 18.B=) ∙ (B;))

19.(3$)=: ∙ 3;$ 20.(5<)); 21.(6=)); 22.(2<)?;);

14




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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Abe:Ijustknowthereissomethinggoingonwiththeselogs.Ijustgraphed-()) = log'(4)).Hereitis:

It’sweirdbecauseIthinkthisgraphisjustatranslationof! = log' ).Isitpossibletheequationofthisgraphcouldbewrittenmorethanoneway?

3. HowwouldyouanswerAbe’squestion?Arethereconditionsthatcouldallowthesamegraphtohavedifferentequations?

Mary:Whenyousay,“atranslationof! = log' )”doyoumeanthatitisjustaverticalorhorizontalshift?Whatcouldthatequationbe?

4. Findanequationfor-())thatshowsittobeahorizontalorverticalshiftof! = log' ).

16




Mary:Iwonderwhytheverticalshiftturnedouttobeup2whenthexwasmultipliedby4.Iwonderifithassomethingtodowiththepowerthatthebaseisraisedto,sincethisisalogfunction.Let’strytoseewhathappenswith! = log'(8))and! = log'(16)).

5. Trytowriteanequivalentequationforeachofthesegraphsthatisaverticalshiftof! = log' ).

a) ! = log'(8)) Equivalentequation:____________________________________________

b)! = log'(16)) Equivalentequation:____________________________________________

17




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.Ifitisnottrueprovideacounter-example.

18




Mary:So,Iwonderifasimilarthinghappensifyouhavedivisioninsidetheargumentofalogfunction.I’mgoingtotrysomeexamples.Ifmytheoryworks,thenallofthesegraphswilljustbeverticalshiftsof! = log' ).

8. HereareAbe’sexamplesandtheirgraphs.TestAbe’stheorybytryingtowriteanequivalentequationforeachofthesegraphsthatisaverticalshiftof! = log' ).

a) ! = log' :;

<= Equivalentequation:_____________________________________________

b) ! = log' :;

>= Equivalentequation:__________________________________________

9. UsetheseexamplestowritearulefordivisioninsidetheargumentofalogthatisliketherulethatMarywroteformultiplicationinsidealog.

19




10. Isthisstatementtrue?Ifitis,givesomeexamplesthatillustratewhyitworks.Ifitisnottrueprovideacounter-example.

Abe:You’redefinitelybrilliantforthinkingofthatmultiplicationrule.ButI’mageniusbecauseI’veusedyourmultiplicationruletocomeupwithapowerrule.Let’ssayyoustartwith:

log'( )?)

Reallythat’sthesameashaving: log'() ∙ ) ∙ ))

So,Icoulduseyourmultiplyingruleandwrite: log' ) + log' ) + log' )Inoticethereare3termsthatareallthesame.Thatmakesit: 3 log' )

Somyruleis: log'()?) = 3 log' )

Ifyourruleistrue,thenIhaveprovenmypowerrule.

Mary:Idon’tthinkit’sreallyapowerruleunlessitworksforanypower.Youonlyshowedhowitmightworkfor3.

Abe:Oh,goodgrief!Ok,I’mgoingtosayitcanbeanynumber),raisedtoanypower,k.Mypowerruleis:

log'()A) = B log' )

Areyousatisfied?

11. ProvideanargumentaboutAbe’spowerrule.Isittrueornot?

20




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




2.3 Chopping Logs – Teacher Notes

A Solidify Understanding Task NotetoTeachers:Accesstographingtechnologyisnecessaryforthistask.Purpose:Thepurposeofthistaskistousestudentunderstandingofloggraphsandlog

expressionstoderivepropertiesoflogarithms.Inthetaskstudentsareaskedtofindequivalent

equationsforgraphsandthentogeneralizethepatternstoestablishtheproduct,quotient,and

powerrulesforlogarithms.

CoreStandardsFocus:

F.IF.8.Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealand

explaindifferentpropertiesofthefunction.

F.LE.4.Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddare


NotetoF.LE.4:Considerextendingthisunittoincludetherelationshipbetweenpropertiesof


andthebasiclogarithmpropertythatlog(xy)=logx+logy.

RelatedStandards:F.BF.5



SMP7–Lookforandmakeuseofstructure

TheTeachingCycle:

Note:Thenatureofthistasksuggestsamoreguidedapproachfromtheteacherthanmanytasks.

Themostproductiveclassroomconfigurationmightbepairs,sothatstudentscaneasilyshifttheir

attentionbackandforthfromwholegroupdiscussiontotheirownwork.

Launch(WholeClass):Beginthetaskbyintroducingtheequation:log'() + +) = log' ) + log' +.

Askstudentswhythismightmakesense.Expecttohearthattheyhave“distributed”thelog.




Withoutjudgingthemeritsofthisidea,askstudentshowtheycouldtesttheclaim.Whentheidea

totestparticularnumberscomesup,setstudentstoworkonquestions#1and#2.Afterstudents

havehadachancetoworkon#2,askastudentthathasanexampleshowingthestatementtobe

untruetosharehis/herwork.Youmayneedtohelprewritethestudent’sworksothatthe

statementsareclearbecausethisisastrategythatstudentswillwanttousethroughoutthetask.

Explore(SmallGroup):Askstudentstoturntheirattentiontoquestion#3.Basedontheirwork

withloggraphsinprevioustasks,theyhavenotseenthegraphofalogfunctionwithamultiplierin

theargument,like-()) = log'(4)).Askstudentshowthe4mightaffectthegraph.BecauseAbe

thinksthefunctionisaverticalshiftof! = log' ),askstudentwhattheequationwouldlooklike

withaverticalshiftsothattheygeneratetheideathattheverticalshifttypicallylookslike

! = 9 + log' ).Askstudentstoinvestigatequestions#3and#4.Asstudentsareworking,lookfora

studentthathasredrawnthex-axisoruseastraightedgetoshowthetranslationofthegraph.

Discuss(WholeClass):Whenstudentshavehadenoughtotimetofindtheverticalshift,havea

studentdemonstratehowtheywereabletotellthatthefunctionisaverticalshiftup2.Itwillhelp

movetheclassforwardinthenextpartofthetaskifastudentdemonstratesredrawingthex-axis

soitiseasytoseethatthegraphisatranslation,butnotadilationof! = log' ).Afterthis

discussion,havestudentsworkonquestions#5,6,and7.

Explore(SmallGroup):Whilestudentsareworking,listenforstudentswhoareabletodescribe

thepatternMaryhasnoticedinthetask.EncouragestudentstotestMary’sconjecturewithsome

numbers,justastheydidinthebeginningofthetask.

Discuss(WholeClass):AskseveralstudentstostateMary’sruleintheirownwords.Tryto

combinethestudentstatementsintosomethinglike:“Thelogofaproductisthesumofthelogs”or

givethemthisstatementandaskthemtodiscusshowitdescribesthepatterntheyhavenoticed.

Haveseveralstudentsshowexamplesthatprovideevidencethatthestatementistrue,butremind

studentsthatafewexamplesdon’tcountasaproof.Afterthisdiscussion,askstudentstocomplete

therestofthetask.

Explore(SmallGroup):Supportstudentsastheyworktorecognizethepatternsandexpressa

rulein#9asbothanequationandinwords.Studentsmayhavedifficultywiththenotation,soask

themtostatetheruleinwordsfirst,andthenhelpthemtowriteitsymbolically.




Discuss(WholeClass):Theremainingdiscussionshouldfolloweachofthequestionsinthetask

from#9onward.Asthediscussionprogresses,showstudentexamplesofeachrule,bothto

provideevidencethattheruleistrueandalsotopracticeusingtherule.Emphasizereasoningthat

helpsstudentstoseethatthelogrulesareliketheexponentrulesbecauseoftherelationship

betweenlogsandexponents.




2.3



READY Topic:RecallingfractionalexponentsWritethefollowingwithanexponent.Simplifywhenpossible.1.√"#

2.√$%& 3.√'() 4.√8+,)

5.√12501#

6.2(8")%) 7.√96() 8.√75",

Rewritewithafractionalexponent.Thenfindtheanswer.9.89:;√3

# =

10.89:%√4

) = 11.89:?√343

# = 12.89:1√3125

# =

SET Topic:UsingthepropertiesoflogarithmstoexpandlogarithmicexpressionsUsethepropertiesoflogarithmstoexpandtheexpressionasasumordifference,and/orconstantmultipleoflogarithms.(Assumeallvariablesarepositive.)13.89:17"

14.89:110A

15.89:1

1

B

16.89:1

C

D

17.89:,"; 18.89:19"% 19.89:%(7")D 20.89:;√'


22



2.3



21.89:1

EFG

H 22.89:1

I√E

F) 23.89:% J

EKLD

E)M 24.89:% J

EK

F#H)M

GO Topic:WritingexpressionsinexponentialformandlogarithmicformConverttologarithmicform.

25.2I = 512 26.10L% = 0.01 27.J%;MLO=

;

%

Writeinexponentialform.

28.89:D2 =

O

% 29.89:Q

)3 = −1 30.89:K

#

(

O%1= 3

23




2.4 Log-Arithm-etic A Practice Understanding Task

AbeandMaryarefeelinggoodabouttheirlogrulesandbraggingabouttheirmathematicalprowesstoalltheirfriendswhenthisexchangeoccurs:

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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24




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 = ________________________________________________

6. log$ 12 = ________________________________________________

25




7. log$ :@A;= ________________________________________________

8. log$ :AB@; = ________________________________________________

9. log$ 486 = ________________________________________________

10. Giventheworkyouhavejustdone,whatothervalueswouldyouneedtofigureoutthevalueofthebase2logforanynumber?

26




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




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.



NotetoF.LE.4:Considerextendingthisunittoincludetherelationshipbetweenpropertiesof


andthebasiclogarithmpropertythatlog(xy)=logx+logy.



SMP6–Attendtoprecision

TheTeachingCycle

Launch(WholeClass):Launchthetaskbyreadingthroughthescenarioandaskingstudentsto

workproblem#1.Followitbyadiscussionoftheiranswers,pointingoutthatequivalentforms

oftenhavedifferentmeaningsinastorycontextandtheycanbehelpfulinsolvingequationsand

graphing.Followthisshortdiscussionbyhavingstudentsworkproblems#2and#3,then




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.




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




2.4



READY Topic:SolvingsimpleexponentialandlogarithmicequationsYouhavesolvedexponentialequationsbeforebasedontheideathat!" = !$, &'!)*+),$&'" = $.

Youcanusethesamelogiconlogarithmicequations.,+.!" = ,+.!$, &'!)*+),$&'" = $

Rewriteeachequationtosetupaone-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.?@6A0= 8

3.?B;A0= 6C

D;

4.34567 −345613 = 0

5.3456(27 − 4) −34568 = 0 6.3456(7 + 2) −345697 = 0

7.IJK 60IJK @;

= 1 8.IJK(L0M@)IJK 6N

= 1 9.IJK L(OPQ)

IJK D6L= 1


28



2.4



SET Topic:RewritinglogsintermsofknownlogsUsepropertiesoflogarithmstorewritetheindicatedlogarithmsintermsofthegivenvalues,

thenfindtheindicatedlogarithmwithusingacalculator.

Donotuseacalculatortoevaluatethelogarithms.

Given:,+.RS ≈ R. U,+.V ≈ W.X,+.Y ≈ W.Z

10.Find345 L

[

11.Find34525

12.Find345 @

6

13.Find34580 14.Find345 @

D;

Given,+.\U ≈ W. S,+.\V ≈ R. V

15.Find345B16

16.Find345B108

17.Find345BB

L^

18.Find345B[

@L 19.Find345B486

20.Find345B18 21.Find345B120 22.Find345BB6

;L

29



2.4



GO Topic:UsingthedefinitionoflogarithmtosolveforxUseyourcalculatorandthedefinitionoflogx(recallthebaseis10)tofindthevalueofx.(Roundyouranswersto4decimals.)

23.logx=-3 24.345x = 1 25.345x = 0

26.345x = @6 27.345x = 1.75 28.345x = −2.2

29.345x = 3.67 30.345x = B;31.345x = 6

30




2.5 Powerful Tens A Practice Understanding Task

TablePuzzles1. Findthemissingvaluesof!inthetables:

c.Whatequationscouldbewritten,intermsof!only,foreachoftherowsthataremissingthe!inthetwotablesabove?

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a.

x " = $%&-2

1100

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

31




f. Whatequationscouldbewritten,intermsof!only,foreachoftherowsthataremissingthe!inthetwotablesabove?

2. Whatstrategydidyouusetofindthesolutionstoequationsgeneratedbythetablesthatcontainedexponentialfunctions?

3. Whatstrategydidyouusetofindthesolutionstoequationsgeneratedbythetablesthatcontainedlogarithmicfunctions?

GraphPuzzles4.Thegraphofy=1023isgivenbelow.Usethegraphtosolvetheequationsforxandlabelthesolutions.

a. 40 = 1023 ! = _____LabelthesolutionwithanAonthegraph.

b. 1023 = 10! = _____LabelthesolutionwithaBonthegraph.

c. 1023 = 0.1! = _____LabelthesolutionwithaConthegraph.

32




5.Thegraphof7 = −2 + log !isgivenbelow.Usethegraphtosolvetheequationsforxandlabelthesolutions.

a. −2 + log ! = −2! = _____

LabelthesolutionwithanAonthegraph.

b. −2 + log ! = 0! = _____

LabelthesolutionwithaBonthegraph.

c.−4 = −2 + log ! ! = _____LabelthesolutionwithaConthegraph.d.−1.3 = −2 + log ! ! = _____LabelthesolutionwithaDonthegraph.e.1 = −2 + log ! ! = _____

6.Arethesolutionsthatyoufoundin#5exactorapproximate?Why?

EquationPuzzlesSolveeachequationfor!

7. 103=10,000 8. 125 = 103 9. 103?@ = 347

10. 5(103?@) = 0.25 11. 10232B = BCD 12. −(103?@) = 16

33




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:





SMP6–Attendtoprecision

TheTeachingCycle:

Launch(WholeClass):

Remindstudentsthattheyareveryfamiliarwithconstructingtablesforvariousfunctions.In

previoustasks,theyhaveselectedvaluesforxandcalculatedthevalueofybaseduponanequation

orotherrepresentation.Theyhavealsoconstructedgraphsbaseduponhavinganequationoraset

ofxandyvalues.Inthistasktheywillbeusingtablesandgraphstoworkinreverse,findingthex

valueforagiveny.





Monitorstudentsastheyworkandlistentotheirstrategiesforfindingthemissingvaluesofx.As

theyareworkingonthetablepuzzles,encouragethemtoconsiderwritingequationsasawayto

tracktheirstrategies.Inthegraphpuzzles,theywillfindtheycanonlygetapproximateanswerson

afewequations.Encouragethemtousethegraphtoestimateavalueandtointerpretthesolution

intheequation.Thepurposeofthetablesandgraphsistohelpstudentsdrawupontheirthinking

fromprevioustaskstosolvetheequations.Remindstudentstoconnecttheideasastheyworkon

theequationpuzzles.


Startthediscussionwithastudentthathaswrittenandsolvedanequationforthethirdrowin

tableb.Theequationwrittenshouldbe:

FG.HI = )($%&)

Askthestudenttodescribehowtheywrotetheequationandthentheirstrategyforsolvingit.Be

suretohavestudentsdescribetheirthinkingabouthowtounwindthefunctionasthestepsare

trackedontheequation.Asktheclasswherethispointwouldbeonagraphofthefunction.Ask

studentswhatthegraphofthefunctionwouldlooklike,andtheyshouldbeabletodescribeabase

10exponentialfunctionwithaverticalstretchof3.

Movethediscussiontotable“e”,focusingonthelastrowofthetable.Again,havestudentswrite

theequation:

) = -./(& + ))

Askthepresentingstudenttodescribehis/herthinkingabouthowtofindthevalueof!inthetable

and,onceagain,trackthestepsalgebraically.Thereareacoupleoflikelymistakesmadeby

studentswhohavetriedtosolvethisequationalgebraically.Iftheyariseduringyourobservation

ofstudents,discussthemhere.Again,connectthesolutiontothegraphofthefunction.Students

shouldbenoticingthatsincelogsandexponentialsareinversefunctions,exponentialequationscan

besolvedwithlogsandlogequationsaresolvedwithexponentials.

Movethediscussiontothegraphof7 = 1023 .Askstudentstodescribehowtheyusedthegraphto

findthesolutionto“a”.Askstudentshowtheycouldcheckthesolutionintheequation.Doesthe




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.




2.5



READY Topic:ComparingthegraphsoftheexponentialandlogarithmicfunctionsThegraphsof!(#) = 10(*+,-(#) = ./- #areshowninthesamecoordinateplane.

Makealistofthecharacteristicsofeachfunction.

1. !(#) = 10(

2.-(#) = ./-#

Eachquestionbelowreferstothegraphsofthefunctions0(1) = 2314567(1) = 8971.State

whethertheyaretrueorfalse.Iftheyarefalse,correctthestatementsothatitistrue.

__________ 3.Everygraphoftheform-(#) = ./- #willcontainthepoint(1,0).

__________ 4.Bothgraphshaveverticalasymptotes.

__________ 5.Thegraphsof!(#)*+,-(#)havethesamerateofchange.

__________ 6.Thefunctionsareinversesofeachother.

__________ 7.Therangeof!(#)isthedomainof-(#).

__________ 8.Thegraphof-(#)willneverreach3.


34



2.5



SET Topic:Solvinglogarithmicequations(base10)bytakingthelogofeachsideEvaluatethefollowinglogarithms9../-10 10../- 10;< 11../- 10<= 12../- 10(

13../->3= 14../-@8;> 15../-BB11>< 16../-CDE

Youcanusethispropertyoflogarithmstohelpyousolvelogarithmicequations.*Note:Thispropertyonlyworkswhenthebaseofthelogarithmmatchesthebaseoftheexponent.Solvetheequationsbyinserting897 onbothsidesoftheequation.(Youwillneedacalculator.)

17.10E = 4.305 18.10E = 0.316 19.10E = 14,52120.10E = 483.059

GO Topic:SolvingequationsinvolvingcompoundinterestFormulaforcompoundinterest:IfPdollarsisdepositedinanaccountpayinganannualrateofinterestrcompounded(paid)ntimesperyear,theaccountwillcontainL = MN2 + P

5Q5Rdollars

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.)

35

Logarithmic Functions - Mathematics Vision Project · Ready, Set, Go Homework: Logarithmic...

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Transcript of Logarithmic Functions - Mathematics Vision Project · Ready, Set, Go Homework: Logarithmic...