Logarithmic Functions - Mathematics Vision Project · Ready, Set, Go Homework: Logarithmic...
Transcript of 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
ALGEBRA II // 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
Thenotationmayseedifferent,butyoucanseetheinversepatternwheretheinputsandoutputsswitch.
Thenextfewproblemswillgiveyouanopportunitytopracticethinkingaboutthispatternandpossiblymakeafewconjecturesaboutotherpatternsrelatedtologarithms.
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Input Output
/(1) = 23 /4-(1) = log)1
1 = 3 3 2" = 8
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
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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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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
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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:____________________________________________________________________________________________________
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ALGEBRA II // 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
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
2.1
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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
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
2.1
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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
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.1
2.1
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GO Topic:ReviewingpropertiesofExponents
Writeeachexpressionasanintegerorasimplefraction.
22.270 23.11(-6)0 24.−3</
25.4<= 26.J
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ALGEBRA II // 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 < $ < 1.
a) &($) = log-$
b) .($) = log/$
c) ℎ($) = log1$
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
d) 2($) = log34$
2. Howdidyoudecidewhatvaluestousefor$inthetable?
3. Howdidyouusethe$valuestofindtheyvaluesinthetable?
4. Whatsimilaritiesdoyouseeinthegraphs?
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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)?
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
• Makeageneralargumentforwhythisistrueforalllogarithmicfunctions.
12. Writeanequationforeachofthefollowingfunctionsthataretransformationsof&($) = log- $.
a)
b)
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
13. Graphandlabeleachofthefollowingfunctions:a) &($) = 2 +log-($ − 1)
b) .($) = −1 + log-($ + 2)
14. Comparethetransformationofthegraphsoflogarithmicfunctionswiththetransformation
ofthegraphsofquadraticfunctions.
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ALGEBRA II // 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 – 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.
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
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.
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
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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.
AlignedReady,Set,Go:LogarithmicFunctions2.2
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
2.2
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READY Topic:SolvingsimplelogarithmicequationsFindtheanswertoeachlogarithmicequation.Thenexplainhoweachequationsupportsthestatement,“Theanswertoalogarithmicequationisalwaystheexponent.”
1. !"#$625 =
2. !"#)243 =
3. !"#$0.2 =
4. !"#.81 =
5. !"#1,000,000 =
6. !"#234 =
SET Topic:ExploringtransformationsonlogarithmicfunctionsAnswerthequestionsabouteachgraph.7.
a. Whatisthevalueofxwhen5(3) = 0?b. Whatisthevalueofxwhen5(3) = 1?c. Whatisthevalueof5(3)when3 = 2?d. Whatwillbethevalueofxwhen5(3) = 4?e. Whatistheequationofthisgraph?
8.
a.Whatisthevalueofxwhen5(3) = 0?b.Whatisthevalueofxwhen5(3) = 1?c.Whatisthevalueof5(3)when3 = 9?d.Whatwillbethevalueofxwhen5(3) = 4?e.Whatistheequationofthisgraph?
READY, SET, GO! Name PeriodDate
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.2
2.2
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9.Usethegraphandthetableofvaluesforthegraphtowritetheequationofthegraph.Explainwhichnumbersinthetablehelpedyouthemosttowritetheequation.
10.Usethegraphandthetableofvaluesforthegraphtowritetheequationofthegraph.Explainwhichnumbersinthetablehelpedyouthemosttowritetheequation.
GO Topic:UsingthepowertoapowerrulewithexponentsSimplifyeachexpression.Answersshouldhaveonlypositiveexponents.11.(2)): 12.(3)); 13.(<))=; 14.(2)>):
15.(?=4)) 16.(@=))=; 17.3; ∙ (3$); 18.B=) ∙ (B;))
19.(3$)=: ∙ 3;$ 20.(5<)); 21.(6=)); 22.(2<)?;);
14
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
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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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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // 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
ALGEBRA II // 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.Ifitisnottrueprovideacounter-example.
18
ALGEBRA II // 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
ALGEBRA II // 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.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
ALGEBRA II // 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
ALGEBRA II // 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:Thepurposeofthistaskistousestudentunderstandingofloggraphsandlog
expressionstoderivepropertiesoflogarithms.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.
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Discuss(WholeClass):Theremainingdiscussionshouldfolloweachofthequestionsinthetask
from#9onward.Asthediscussionprogresses,showstudentexamplesofeachrule,bothto
provideevidencethattheruleistrueandalsotopracticeusingtherule.Emphasizereasoningthat
helpsstudentstoseethatthelogrulesareliketheexponentrulesbecauseoftherelationship
betweenlogsandexponents.
AlignedReady,Set,Go:LogarithmicFunctions2.3
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
READY Topic:RecallingfractionalexponentsWritethefollowingwithanexponent.Simplifywhenpossible.1.√"#
2.√$%& 3.√'() 4.√8+,)
5.√12501#
6.2(8")%) 7.√96() 8.√75",
Rewritewithafractionalexponent.Thenfindtheanswer.9.89:;√3
# =
10.89:%√4
) = 11.89:?√343
# = 12.89:1√3125
# =
SET Topic:UsingthepropertiesoflogarithmstoexpandlogarithmicexpressionsUsethepropertiesoflogarithmstoexpandtheexpressionasasumordifference,and/orconstantmultipleoflogarithms.(Assumeallvariablesarepositive.)13.89:17"
14.89:110A
15.89:1
1
B
16.89:1
C
D
17.89:,"; 18.89:19"% 19.89:%(7")D 20.89:;√'
READY, SET, GO! Name PeriodDate
22
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.3
2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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21.89:1
EFG
H 22.89:1
I√E
F) 23.89:% J
EKLD
E)M 24.89:% J
EK
F#H)M
GO Topic:WritingexpressionsinexponentialformandlogarithmicformConverttologarithmicform.
25.2I = 512 26.10L% = 0.01 27.J%;MLO=
;
%
Writeinexponentialform.
28.89:D2 =
O
% 29.89:Q
)3 = −1 30.89:K
#
(
O%1= 3
23
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.4 Log-Arithm-etic A Practice Understanding Task
AbeandMaryarefeelinggoodabouttheirlogrulesandbraggingabouttheirmathematicalprowesstoalltheirfriendswhenthisexchangeoccurs:
Stephen:Iguessyouthinkyou’reprettysmartbecauseyoufiguredoutsomelogrules,butIwanttoknowwhatthey’regoodfor.
Abe:Well,we’veseenalotoftimeswhenequivalentexpressionsarehandy.Sometimeswhenyouwriteanexpressionwithavariableinitinadifferentwayitmeanssomethingdifferent.
1. Whataresomeexamplesfromyourpreviousexperiencewhereequivalentexpressionswereuseful?
Mary:IwasthinkingabouttheLogLogictaskwhereweweretryingtoestimateandordercertainlogvalues.Iwaswonderingifwecoulduseourlogrulestotakevaluesweknowandusethemtofindvaluesthatwedon’tknow.
Forinstance:Let’ssayyouwanttocalculatelog$ 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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tps:/
/flic
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9FaH
24
ALGEBRA II // 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 = ________________________________________________
6. log$ 12 = ________________________________________________
25
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7. log$ :@A;= ________________________________________________
8. log$ :AB@; = ________________________________________________
9. log$ 486 = ________________________________________________
10. Giventheworkyouhavejustdone,whatothervalueswouldyouneedtofigureoutthevalueofthebase2logforanynumber?
26
ALGEBRA II // 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
ALGEBRA II // 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
ALGEBRA II // 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.
ALGEBRA II // 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
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
2.4
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Needhelp?Visitwww.rsgsupport.org
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
READY, SET, GO! Name PeriodDate
28
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
2.4
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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
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.4
2.4
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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
ALGEBRA II // 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. Findthemissingvaluesof!inthetables:
c.Whatequationscouldbewritten,intermsof!only,foreachoftherowsthataremissingthe!inthetwotablesabove?
CC B
Y El
i Chr
istm
an
http
s://f
lic.k
r/p/
avCd
Hc
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
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // 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 – 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.
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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!inthetable
and,onceagain,trackthestepsalgebraically.Thereareacoupleoflikelymistakesmadeby
studentswhohavetriedtosolvethisequationalgebraically.Iftheyariseduringyourobservation
ofstudents,discussthemhere.Again,connectthesolutiontothegraphofthefunction.Students
shouldbenoticingthatsincelogsandexponentialsareinversefunctions,exponentialequationscan
besolvedwithlogsandlogequationsaresolvedwithexponentials.
Movethediscussiontothegraphof7 = 1023 .Askstudentstodescribehowtheyusedthegraphto
findthesolutionto“a”.Askstudentshowtheycouldcheckthesolutionintheequation.Doesthe
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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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.
READY, SET, GO! Name PeriodDate
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ALGEBRA II // MODULE 2
LOGARITHMIC FUNCTIONS – 2.5
2.5
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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.)
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