Logarithmic FunctionsSECONDARY MATH III // MODULE 2 LOGARITHMIC FUNCTIONS Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 2

Logarithmic Functions

SECONDARY

MATH THREE

An Integrated Approach

SECONDARY MATH III // MODULE 2

LOGARITHMIC FUNCTIONS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 2 - TABLE OF CONTENTS

LOGARITHMIC FUNCTIONS

2.1 Log Logic – A Develop Understanding Task Evaluate and compare logarithmic expressions. (F.BF.5, F.LE.4) Ready, Set, Go Homework: Logarithmic Functions 2.1

2.2 Falling Off a Log – A Solidify Understanding Task Graph logarithmic functions with transformations (F.BF.3, F.BF.5, F.IF.7e) Ready, Set, Go Homework: Logarithmic Functions 2.2

2.3 Chopping Logs – A Solidify Understanding Task Explore properties of logarithms. (F.IF.8, F.LE.4) Ready, Set, Go Homework: Logarithmic Functions 2.3

2.4 Log-Arithm-etic – A Practice Understanding Task Use log properties to evaluate expressions (F.IF.8, F.LE.4) Ready, Set, Go Homework: Logarithmic Functions 2.4

2.5 Powerful Tens – A Practice Understanding Task Solve exponential and logarithmic functions in base 10 using technology. (F.LE.4) Ready, Set, Go Homework: Logarithmic Functions 2.5


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

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

1 = 3 3 2" = 8

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Thenotationmayseedifferent,butyoucanseetheinversepatternwheretheinputsandoutputsswitch.

Thenextfewproblemswillgiveyouanopportunitytopracticethinkingaboutthispatternandpossiblymakeafewconjecturesaboutotherpatternsrelatedtologarithms.

Placethefollowingexpressionsonthenumberline.Usethespacebelowthenumberlinetoexplainhowyouknewwheretoplaceeachexpression.

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

Explain:____________________________________________________________________________________________________

2. A.log"81 B.log-.100 C.log78D.log825 E.log)32

Explain:____________________________________________________________________________________________________

3. A.log:7 B.log69 C.log--1D.log-.1

Explain:____________________________________________________________________________________________________

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

Name Date

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

0 1 2 3 4 5TheMathWorksheetSite.com

Name Date

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


2




4. A.log) <-=> B.log-. <-

-...>C.log8 <--)8>D.log? <

-?>

Explain:____________________________________________________________________________________________________

5. A.log=16 B.log)16C.log716D.log-?16

Explain:____________________________________________________________________________________________________

6. A.log)5 B.log810 C.log?1 D.log85 E.log-.5

Explain:____________________________________________________________________________________________________

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

Explain:____________________________________________________________________________________________________

8. A.log"3) B.log854) C.log?6. D.log=44- E.log)2"

Explain:____________________________________________________________________________________________________

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


Name Date

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


Name Date

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


Name Date

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


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


3




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

9. ThevalueoflogB 1ispositive.

Explain:____________________________________________________________________________________________________

10. logB 1isnotavalidexpressionifxisanegativenumber.

Explain:____________________________________________________________________________________________________

11. logB 1 = 0foranybase,b>0.

Explain:____________________________________________________________________________________________________

12. logB C = 1foranyb>0.

Explain:____________________________________________________________________________________________________

13. log) 1<log" 1foranyvalueofx.

Explain:____________________________________________________________________________________________________

14. logDCE = Fforanyb>0.

Explain:____________________________________________________________________________________________________

4


LOGRITHMIC FUNCTIONS –


2.1

Need$help?$$Visit$www.rsgsupport.org$

REA DY Topic:**Graphing*exponential*equations**Graph$each$function$over$the$domain$ −! ≤ ! ≤ ! .$*1.**! = 2!!!!*

*2.**! = 2 ∙ 2! *

*

3.**! = !!!*

*

4.**! = 2 !!!*

*

* * * ***

5.***Compare*graph*#1*to*graph*#2.$Multiplying*by*2*should*generate*a*dilation*of*the*graph,*but*the*graph*looks*like*it*has*been*translated*vertically.**How*do*you*explain*that?*

*

*

6.***Compare*graph*#3*to*graph*#4.$$Is*your*explanation*in*#5*still*valid*for*these*two*graphs?*Explain.*

SET Topic:**Writing*the*logarithmic*form*of*an*exponential*equation.*

Definition$of$Logarithm:$$$$For$all$positive$numbers$a,$where$a$≠$1,$and$all$positive$numbers$x,$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! = !"#!!$means$the$same$as$x"="!!.$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$(Note$the$base$of$the$exponent$and$the$base$of$the$logarithm$are$both$a.)$

$

–5 5

20

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8

6

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2

–2–5 5

20

18

16

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8

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–2–5 5

20

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8

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2

–2–5 5

20

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16

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10

8

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4

2

–2

READY, SET, GO! ******Name* ******Period***********************Date*

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2.1


7.***Why*is*it*important*that*the*definition*of*logarithm*states*that*the*base*of*the*logarithm*does*not*equal*1?**

*8.***Why*is*it*important*that*the*definition*states*that*the*base*of*the*logarithm*is*positive?*

*

9.***Why*is*it*necessary*that*the*definition*states*that*x*in*the*expression*!"#!!**is*positive?**

Write$the$following$exponential$equations$in$logarithmic$form.$

Exponential$form$ Logarithmic$form$ Exponential$form$ Logarithmic$form$

*10.**5! = 625**

* *11.**3! = 9*

*

12.**!!!!= 8* * 13.**4!! = !

!"**

*14.***10! = 10000*

* *15.**!!*=*x*

*

*

16.**Compare*the*exponential*form*of*an*equation*to*the*logarithmic*form*of*an*equation.**What*part*of*the*exponential*equation*is*the*answer*to*the*logarithmic*equation?*

Topic:**Considering*values*of*logarithmic*functions**Answer$the$following$questions.*If$yes,$give$an$example$or$the$answer.$$If$no,$explain$why$not.$

17.**Is*it*possible*for*a*logarithm*to*equal*a*negative*number?**

18.**Is*it*possible*for*a*logarithm*to*equal*zero?*

19.**Does*!"#!0*have*an*answer?*

20.**Does*!"#!1*have*an*answer?*

21.**Does*!!"!!!*have*an*answer?$

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2.1


GO Topic:**Reviewing*properties*of*Exponents*

Write$each$expression$as$an$integer$or$a$simple$fraction.*

22.***270* 23.***11(Y6)0* 24.***−3!!*

*

25.****4!!* 26.*****!!!!* * 27.*****

!!!!*

*

28.****3 !"!!!!

!* 29.*****

!!!!* * 30.*****

!"!!!!! *

*

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2.2 Falling Off a Log A Solidify Understanding Task

1. Constructatableofvaluesandagraphforeachofthefollowingfunctions.Besuretoselectatleasttwovaluesintheinterval0<x<1.

a) !(#) = log)#

b) *(#) = log+#

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c) ℎ(#) = log-#

d) .(#) = log/0#

2. Howdidyoudecidewhatvaluestouseforxinyourtable?

3. Howdidyouusethexvaluestofindtheyvaluesinthetable?

9




4. Whatsimilaritiesdoyouseeinthegraphs?

5. Whatdifferencesdoyouobserveinthegraphs?

6. Whatistheeffectofchangingthebaseonthegraphofalogarithmicfunction?

Let’sfocusnowon.(#) = log/0#sothatwecanusetechnologytoobservetheeffectsofchangingparametersonthefunction.Becausebase10isaverycommonlyusedbaseforexponentialandlogarithmicfunctions,itisoftenabbreviatedandwrittenwithoutthebase,likethis:1(2) = 3452.

7. Usetechnologytograph7 = log #.Howdoesthegraphcomparetothegraphthatyouconstructed?

8. Howdoyoupredictthatthegraphof7 = 8 + log #willbedifferentfromthegraphof7 = log #?

9. Testyourpredictionbygraphing7 = 8 + log #forvariousvaluesofa.Whatistheeffectofaonthegraph?Makeageneralargumentforwhythiswouldbetrueforalllogarithmicfunctions.

10. Howdoyoupredictthatthegraphof7 = log(# + :)willbedifferentfromthegraphof7 = log #?

10




11. Testyourpredictionbygraphing7 = log(# + :)forvariousvaluesofb.• Whatistheeffectofaddingb?

• Whatwillbetheeffectofsubtractingb(oraddinganegativenumber)?

• Makeageneralargumentforwhythisistrueforalllogarithmicfunctions.

12. Writeanequationforeachofthefollowingfunctionsthataretransformationsof!(#) = log) #.

a.

b.

11




13. Graphandlabeleachofthefollowingfunctions:a. !(#) = 2 +log)(# − 1)

b. *(#) = −1 + log)(# + 2)

14. Comparethetransformationofthegraphsoflogarithmicfunctionswiththetransformation

ofthegraphsofquadraticfunctions.

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2.2


REA DY Topic:**Solving*simple*logarithmic*equations**Find$the$answer$to$each$logarithmic$equation.$$Then$explain$how$each$equation$supports$the$statement,$“The%answer%to%a%logarithmic%equation%is%always%the%exponent.”%$

1. !"#!625 =%

*

2. !"#!243 =*

*

3. !"#!0.2 =*

*

4. !"#!81 =*

*

5. !"#1,000,000 =*

*

6. !"#!!! =*

*

SET Topic:**Exploring*transformations*on*logarithmic*functions**Answer$the$questions$about$each$graph.**7.***

*a. What*is*the*value*of*x*when*! ! = 0?*b. What*is*the*value*of*x*when*! ! = 1?*c. What*is*the*value*of*! ! *when*! = 2?*d. What*will*be*the*value*of*x*when*! ! = 4?*e. What*is*the*equation*of*this*graph?*

*8.***

*a.*****What*is*the*value*of*x*when*! ! = 0?***************b.*****What*is*the*value*of*x*when*! ! = 1?*******************c.******What*is*the*value*of*! ! *when*! = 9?***************d.******What*will*be*the*value*of*x*when*! ! = 4?*****e.******What*is*the*equation*of*this*graph?*

***


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2.2


9.**Use*the*graph*and*the*table*of*

values*for*the*graph*to*write*the*

equation*of*the*graph.*

Explain*which*numbers*in*the*

table*helped*you*the*most*to*

write*the*equation.*

*

*

*

10.**Use*the*graph*and*the*table*of*

values*for*the*graph*to*write*the*

equation*of*the*graph.*

Explain*which*numbers*in*the*

table*helped*you*the*most*to*

write*the*equation.*

*

*

* GO Topic:**Using*the*power*to*a*power*rule*with*exponents**Simplify$each$expression.$$Answers$should$have$only$positive$exponents.$$11.*** 2! !***** * 12.*** !! !*** * 13.*** !! !!***** * 14.*** 2!! !*****

*

*

*

15.*** !!! !********* 16.***** !!! !!******** 17.****!! ∙ !! !***** 18.***!!! ∙ !! !*

*

*

*

19.*** !! !! ∙ !!"* 20.*** 5!! !***** * 21.*** 6!! !***** * 22.*** 2!!!! !*********

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

It’sweirdbecauseIthinkthatthisgraphisjustatranslationof! = log' ).Isitpossiblethattheequationofthisgraphcouldbewrittenmorethanoneway?

3. HowwouldyouanswerAbe’squestion?Arethereconditionsthatcouldallowthesamegraphtohavedifferentequations?

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

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

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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.Ifitisnottrueprovideacounterexample.

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

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

a) ! = log' :;

<= Equivalentequation:_____________________________________________

b) ! = log' :;

>= Equivalentequation:__________________________________________

9. UsetheseexamplestowritearulefordivisioninsidetheargumentofalogthatisliketherulethatMarywroteformultiplicationinsidealog.

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10. Isthisstatementtrue?Ifitis,givesomeexamplesthatillustratewhyitworks.Ifitisnottrueprovideacounterexample.

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

log'( )?)

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

So,Icoulduseyourmultiplyingruleandwrite: log' ) + log' ) + log' )Inoticethatthereare3termsthatareallthesame.Thatmakesit: 3 log' )

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

Ifyourruleistrue,thenIhaveprovenmypowerrule.

Mary:Idon’tthinkit’sreallyapowerruleunlessitworksforanypower.Youonlyshowedhowitmightworkfor3.

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

log'()A) = B log' )

Areyousatisfied?

11. ProvideanargumentaboutAbe’spowerrule.Isittrueornot?

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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?

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2.3


REA DY Topic:**Recalling*fractional*exponents**Write$the$following$with$an$exponent.$$Simplify$when$possible.$$1.*** !! *

*

2.*** !!! * 3.*** !!! * 4.*** 8!!! *

5.*** 125!!!

*

6.*** 8! !! * 7.*** 9!!! * 8.*** 75!!*

$Rewrite$with$a$fractional$exponent.$$Then$find$the$answer.$*9.***!"#! 3! = **

*

*10.***!"#! 4! = *

*11.***!"#! 343! = *

*12.***!"#! 3125! = *

SET Topic:**Using*the*properties*of*logarithms*to*expand*logarithmic*expressions**Use$the$properties$of$logarithms$to$expand$the$expression$as$a$sum$or$difference,$and/or$constant$multiple$of$logarithms.$$(Assume*all*variables*are*positive.)***13.***!"#!7!*

*14.***!"#!10!*

*15.***!"#! !!*

*16.***!"#! !!*****

17.***!"#!!! * 18.***!"#!9!!* 19.***!"#! 7! !* 20.***!"#! !******


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2.3


*21.***!"#! !"#! * 22.***!"#! ! !

!! * 23.***!"#! !!!!!! * 24.***!"#! !!

!!!! *******

GO Topic:**Writing*expressions*in*exponential*form*and*logarithmic*form**Convert$to$logarithmic$form.$$

25.***2! = 512* * * 26.***10!! = 0.01*** * 27.*** !!!!= !

!*********Write$in$exponential$form.$

*28.***!"#!2 = ! !!**** * 29.***!"#!

!3 = −1***** * 30.***!"#!

!

!!"# = 3*

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2.4 Log-Arithm-etic A Practice Understanding Task

AbeandMaryarefeelinggoodabouttheirlogrulesandbraggingabouttheirmathematicalprowesstoalloftheirfriendswhenthisexchangeoccurs:

Stephen:Iguessyouthinkyou’reprettysmartbecauseyoufiguredoutsomelogrules,butIwanttoknowwhatthey’regoodfor.

Abe:Well,we’veseenalotoftimeswhenequivalentexpressionsarehandy.Sometimeswhenyouwriteanexpressionwithavariableinitinadifferentwayitmeanssomethingdifferent.

1. Whataresomeexamplesfromyourpreviousexperiencewhereequivalentexpressionswereuseful?

Mary:IwasthinkingabouttheLogLogictaskwhereweweretryingtoestimateandordercertainlogvalues.Iwaswonderingifwecoulduseourlogrulestotakevaluesweknowandusethemtofindvaluesthatwedon’tknow.

Forinstance:Let’ssayyouwanttocalculatelog$ 6.Ifyouknowwhatlog$ 2andlog$ 3arethenyoucanusetheproductruleandsay:

log$(2 ∙ 3) = log$ 2 + log$ 3

Stephen:That’sgreat.Everyoneknowsthatlog$ 2 = 1,butwhatislog$ 3?

Abe:Oh,Isawthissomewhere.Uh,log$ 3 =1.585.SoMary’sideareallyworks.Yousay:

log$(2 ∙ 3) = log$ 2 + log$ 3

= 1 + 1.585

= 2.585

log$ 6=2.585

2. Basedonwhatyouknowaboutlogarithms,explainwhy2.585isareasonablevalueforlog$ 6.

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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 = ________________________________________________

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6. log$ 12 = ________________________________________________

7. log$ :@A;= ________________________________________________

8. log$ :AB@; = ________________________________________________

9. log$ 486 = ________________________________________________

10. Giventheworkthatyouhavejustdone,whatothervalueswouldyouneedtofigureoutthevalueofthebase2logforanynumber?

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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

_____________________________________________________

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2.4

Needhelp?Visitwww.rsgsupport.org

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

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

Rewriteeachequationsothatyousetupaone-to-onecorrespondencebetweenalloftheparts.Thensolveforx.Example:Originalequation

a.)30 = 81

b.)34567 − 34565 = 0

Rewrittenequation:

30 = 3;

34567 = 34565

Solution:

7 = 4

7 = 5

1.30=; = 243

2. ?6

0= 8

3. @;

0=

6A

B;

4.34567 − 345613 = 0

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

7.EFG 60EFG ?;

= 1 8.EFG H0I?EFG 6J

= 1 9.EFG HKLM

EFG B6H= 1

READY, SET, GO! Name PeriodDate

28




2.4


SET Topic:RewritinglogsintermsofknownlogsUsethegivenvaluesandthepropertiesoflogarithmstofindtheindicatedlogarithm.

Donotuseacalculatortoevaluatethelogarithms.

Given:NOP QR ≈ Q. TNOP U ≈ V. WNOP X ≈ V. Y

10.Findlog H]

11.Findlog 25

12.Findlog ?6

13.Findlog 80 14.Findlog ?

B;

Given,+.^T ≈ V. R,+.^U ≈ Q. U

15.Find345@16

16.Find345@108

17.Find345@@

H`

18.Find345@]

?H 19.Find345@486

20.Find345@18 21.Find345@120 22.Find345@@6

;H

29




2.4


GO Topic:Usingthedefinitionoflogarithmtosolveforx.Useyourcalculatorandthedefinitionoflogx(recallthebaseis10)tofindthevalueofx.(Roundyouranswersto4decimals.)

23.logx=-3 24.logx = 1 25.logx = 0

26.logx = ?6 27.logx = 1.75 28.logx = −2.2

29.logx = 3.67 30.logx = @;31.logx = 6

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2.5 Powerful Tens A Practice Understanding Task

TablePuzzles1. Usethetablestofindthemissingvaluesofx:

c.Whatequationscouldbewritten,intermsofxonly,foreachoftherowsthataremissingthexinthetwotablesabove?

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

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f.Whatequationscouldbewritten,intermsofxonly,foreachoftherowsthataremissingthexinthetwotablesabove?

2. Whatstrategydidyouusetofindthesolutionstoequationsgeneratedbythetablesthatcontainedexponentialfunctions?

3. Whatstrategydidyouusetofindthesolutionstoequationsgeneratedbythetablesthatcontainedlogarithmicfunctions?

GraphPuzzles4.Thegraphofy=1012 isgivenbelow.Usethegraphtosolvetheequationsforxandlabelthesolutions.

a. 40 = 1012 4 = _____LabelthesolutionwithanAonthegraph.

b. 1012 = 104 = _____LabelthesolutionwithaBonthegraph.

c. 1012 = 0.14 = _____LabelthesolutionwithaConthegraph.

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5.Thegraphofy= − 2 + log 4isgivenbelow.Usethegraphtosolvetheequationsforxandlabelthesolutions.

a. −2 + log 4 = −24 = _____

LabelthesolutionwithanAonthegraph.

b. −2 + log 4 = 04 = _____

LabelthesolutionwithaBonthegraph.

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

6.Arethesolutionsthatyoufoundin#5exactorapproximate?Why?

EquationPuzzles:Solveeachequationforx:

7. 102=10,000 8. 125 = 102 9. 102>? = 347

10. 5(102>?) = 0.25 11. 10121A = ABC 12. −(102>?) = 16

33




2.5


READY Topic:ComparingthegraphsoftheexponentialandlogarithmicfunctionsThegraphsof! " = 10&()*+ " = log "areshowninthesamecoordinateplane.

Makealistofthecharacteristicsofeachfunction.

1. ! " = 10&

2.+ " = log "

Eachquestionbelowreferstothegraphsofthefunctions/ 0 = 1203456 0 = 789 0.State

whethertheyaretrueorfalse.Iftheyarefalse,correctthestatementsothatitistrue.

__________ 3.Everygraphoftheform+ " = log "willcontainthepoint(1,0).

__________ 4.Bothgraphshaveverticalasymptotes.

__________ 5.Thegraphsof! " ()*+ " havethesamerateofchange.

__________ 6.Thefunctionsareinversesofeachother.

__________ 7.Therangeof! " isthedomainof+ " .

__________ 8.Thegraphof+ " willneverreach3.

READY, SET, GO! Name PeriodDate

34




2.5


SET Topic:Solvinglogarithmicequations(base10)bytakingthelogofeachsideEvaluatethefollowinglogarithms9.log 10 10.log 10;< 11.log 10<= 12.log 10&

13.>?+@3= 14.>?+B8;@ 15.>?+DD11@< 16.>?+EFG

Youcanusethispropertyoflogarithmstohelpyousolvelogarithmicequations.

*Note:Thispropertyonlyworkswhenthebaseofthelogarithmmatchesthebaseoftheexponent.

SolvetheequationsbyinsertingHI64onbothsidesoftheequation.(Youwillneedacalculator.)

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

GO Topic:SolvingequationsinvolvingcompoundinterestFormulaforcompoundinterest:IfPdollarsisdepositedinanaccountpayinganannualrateof

interestrcompounded(paid)ntimesperyear,theaccountwillcontainP = Q 1 + S4

4Tdollars

aftertyears.21.Howmuchmoneywilltherebeinanaccountattheendof10yearsif$3000isdepositedat6%annualinterestcompoundedasfollows: (Assumenowithdrawalsaremade.) a.) annually b.) semiannually c.) quarterly d.) daily(Usen=365.)22.Findtheamountofmoneyinanaccountafter12yearsif$5,000isdepositedat7.5%annualinterestcompoundedasfollows: (Assumenowithdrawalsaremade.) a.) annually b.) semiannually c.) quarterly d.) daily(Usen=365.)

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Logarithmic FunctionsSECONDARY MATH III // MODULE 2 LOGARITHMIC FUNCTIONS Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org