IMPActivity:ComparingExponentialandLogarithmicRules 1 !

Comparing Exponential and Logarithmic Rules

Task1:Lookingcloselyatexponentialandlogarithmicpatterns…1)Inapriorlessonyougraphedandthencomparedanexponentialfunctionwithalogarithmic

functionandfoundthatthefunctionsare__________________________functions.

2)Whenafunctionistheinverseofanotherfunctionweknowthatifthe_______________of

onefunctionmapsontotheoutputofanotherfunctionthentheinversemapsthe

_________________tothe________________.

3)Giventhefunctionf(x)=2xandatablewithcorrespondingvalues

x -2 -1 0 1 2 3

f(x) 14

12

1 2 4 8

completethetablefortheinversefunction,𝑓!!(𝑥)=𝑙𝑜𝑔!𝑥.

x

𝑓!!(𝑥)

4)Usingthetables,if2x=8,thenx=______________.If𝑙𝑜𝑔!𝑦=3,theny=______________.5)Describewhatyounoticedabouttherelationshipbetweentheexponentialandlogarithmicequationsinnumber4.___________________________________________________________________________________________________________________________________________________________6)Oncemore,if2x=2,thenx=______________.If𝑙𝑜𝑔!𝑦=1,theny=______________.

7)Simplifythefollowing:

a)𝑙𝑜𝑔!4 =______ b)𝑙𝑜𝑔!27=______ c)𝑙𝑜𝑔!16=______

Name_____________________________________Date_____________Period___________

Let’smakearuleforthisrelationship…..

If𝒚 = 𝒃𝒙,then𝒍𝒐𝒈𝒃______=_______.

Weread𝒍𝒐𝒈𝒃_____=______as__________________________________________________.

IMPActivity:ComparingExponentialandLogarithmicRules 2 !

Task2:Usingsomeexamplestodiscoveraloglaw…Firstmultiply4by8,thenfindthelog.

log! 4 ∙ 8

log! ________

_________

Comparethistofindingthelogofeachfactorseparately:log! 4 =__________andlog! 8 =__________

Multiply10by1,000,000,thenfindthelog.

log!" 10 ∙ 1,000,000

log!" ____________________

_________

Comparethistofindingthelogofeachfactorseparately:log!" 10 =______andlog!" 1,000,000 =______

Multiplyn4byn6,thenfindthelog.

log! 𝑛! ∙ 𝑛!

log! _________

_________

Comparethistofindingthelogofeachfactorseparately:log! 𝑛! =______andlog! 𝑛! =______

Remembertheexponentruleformultiplyingwiththesamebase:

𝒏𝒙 ∙ 𝒏𝒚 = 𝒏𝒙!𝒚

Writethematchingruleforlogarithms:

𝐥𝐨𝐠 𝒂 ∙ 𝒃 = 𝐥𝐨𝐠𝒂 _____ 𝐥𝐨𝐠𝒃

ThisisknownastheLogarithmicProductRule.

8)Explainwhateachoftheserulesmeansusingcompletesentences.__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________9)Makeoneexampleofyourownforeachoftheselaws._____________________________________________________________________

IMPActivity:ComparingExponentialandLogarithmicRules 3 !

Task3:Usingsomeotherexamplestodiscoverasecondloglaw…Expand4tothe3rdpower,thenusetherule

fromthepreviouspagetofindthelog.

log! 4!

log! __________

_________

Comparethistofindingthelogof4.log! 4 =__________Howcanyouusethe3?

Expand1,000tothe3rdpower,thenusethe

rulefromthepreviouspagetofindthelog.

log!" 1,000!

log!" _________________________

_________

Comparethistofindingthelogof1,000log!" 1000 =______Howcanyouusethe3?

Expandn4tothe5thpower,thenusetherule

fromthepreviouspagetofindthelog.

log! 𝑛! !

log! __________________________

_________

Comparethistofindingthelogof𝑛!.log! 𝑛! =______Howcanyouusethe5?

Remembertheexponentruleforraisingapowertoapower:

𝒏𝒙 𝒚 = 𝒏𝒙𝒚

Writethematchingruleforlogarithms:

𝐥𝐨𝐠 𝒂𝒃 = ___________________

ThisisknownastheLogarithmicPowerRule.10)Explainwhateachoftherulesmeansusingcompletesentences.__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

IMPActivity:ComparingExponentialandLogarithmicRules 4 !

____________________________________________________________________________________________________________________________________________________________11)Makeoneexampleofyourownforeachoftheselaws._________________________________________________________________________12)Usingwhatyouhavelearnedabouttheproductruleforlogs,iflog 𝑎 ∙ 𝑏 = log𝑎 + log 𝑏,thenhowmightwebeabletorewritelog !

!?

________________________________

ThisisknownastheLogarithmicQuotientRule.

IMPActivity:ComparingExponentialandLogarithmicRules TeacherDirections5 !

Teacher Directions: Comparing Exponential and Logarithmic Rules

MaterialsCalculatorObjectiveStudentswillexaminespecificexamplesoflogcomputationstodiscoveranddescribetwobasiclogrulesandcomparethemtorelatedexponentialrules.DirectionsHavestudentsgetintogroupsandpassouttheactivitysheet.Remindstudentsaboutwhattheylearnedaboutthegraphsoflogarithmicandexponentialfunctions,thattheyareinversefunctions.Letstudentsknowthattheywillnowbelookingforpatternsbetweenlogarithmicexpressionsandexpressionswithexponentsinordertocometosomegeneralrules.Inthislesson,wholeclassdiscussionshouldcomeafterstudentexplorationandeffortatseeingthepatternandwritingtherule.Somepairsorteamsofstudentswillrecognizethepatternfasterthanothers.Youwillneedtodecidewhentohaveawholeclassdiscussionaboutthepatternandtherule.Thisshouldnotbeimmediatelyafterthefastestteammakesthediscovery,butitmaynotbepracticaltowaituntileveryteamhas.Task1:Lookingcloselyatexponentialandlogarithmicpatterns…ForTask1,itmaybehelpfulforstudentstohaveouttheExponentialandLogarithmicGraphslessonsothattheycanlookmorecloselyatthepatternsbetweenthetablesofanexponentialfunctionanditsinverse,thelogfunction.Havestudentscompletequestions1and2withtheirgroupandthenrandomlyselectagrouptosharetheiranswers.

1)Inapriorlessonyougraphedandthencomparedanexponentialfunctionwithalogarithmicfunctionandfoundthatthefunctionsareinversefunctions.

2)Whenafunctionistheinverseofanotherfunctionweknowthatiftheinputofonefunctionmapsontotheoutputofanotherfunctionthentheinversemapstheoutputtotheinput.

Nexthavestudentscompletequestions3,4and5withtheirgroupsandonceagain,randomlyselectagrouptosharetheiranswersandtherelationshiporpatterntheysaw.Recordanswerstonumber5ontheboard;youmayneedtocallonafewgroupsuntilyougetthepatternwearelookingfor.

IMPActivity:ComparingExponentialandLogarithmicRules TeacherDirections6 !

3)Giventhefunctionf(x)=2xandatablewithcorrespondingvalues

x -2 -1 0 1 2 3

f(x) 14

12

1 2 4 8

completethetablefortheinversefunction,𝑓!!(𝑥)=𝑙𝑜𝑔!𝑥.

x 14

12

1 2 4 8

𝑓!!(𝑥) -2 -1 0 1 2 3

4)Usingthetables,if2x=8,thenx=3.If𝑙𝑜𝑔!𝑦 =3,theny=8.5)Describewhatyounoticedabouttherelationshipbetweentheexponentialandlogarithmicequationsinnumber4.Studentanswersmayvary,butshouldbesomethingsuchas:Inoticedthatthebaseoftheexponentialequationisthebaseofthelogarithmicequation.IntheexponentialequationifIwanttoknowwhatxisthink“twotowhatpowergivesmeeight?”.

Havestudentsworkonproblem6andthenwriteintheruleasaclass.

6)Oncemore,if2x=2,thenx=1.If𝑙𝑜𝑔!𝑦=1,theny=2.

Havestudentscompletenumber7inpreparationforTask2.Ifstudentsneedafewmoreexamplesbeforemovingon,providethemwithafewmore,usingnumbersfromthetablesabove.

7)Simplifythefollowing:a)𝑙𝑜𝑔!4 =2 b)𝑙𝑜𝑔!27=3 c)𝑙𝑜𝑔!16=2

Let’smakearuleforthisrelationship…..

If𝒚 = 𝒃𝒙,then𝒍𝒐𝒈𝒃 y=x.

Weread𝒍𝒐𝒈𝒃y=xaslogbasebofyequalsx.

IMPActivity:ComparingExponentialandLogarithmicRules TeacherDirections7 !

Task2:Usingsomeexamplestodiscoveraloglaw…Onceagain,letstudentsknowthattheywillnowbelookingforinordertocometosomegeneralrules.Inthislesson,wholeclassdiscussionshouldcomeafterstudentexplorationandeffortatseeingthepatternandwritingtherule.Somepairsorteamsofstudentswillrecognizethepatternfasterthanothers.Youwillneedtodecidewhentohaveawholeclassdiscussionaboutthepatternandtherule.Thisshouldnotbeimmediatelyafterthefastestteammakesthediscovery,butitmaynotbepracticaltowaituntileveryteamhas.Firstmultiply4by8,thenfindthelog.

log! 4 ∙ 8

log! 32

5

Comparethistofindingthelogofeachfactorseparately:log! 4 =2andlog! 8 =3

Multiply10by1,000,000,thenfindthelog.

log!" 10 ∙ 1,000,000

log!" 10,000,000

7

Comparethistofindingthelogofeachfactorseparately:log!" 10 =1andlog!" 1,000,000 =6

Multiplyn4byn6,thenfindthelog.

log! 𝑛! ∙ 𝑛!

log! 𝑛!"

10

Comparethistofindingthelogofeachfactorseparately:log! 𝑛! =4andlog! 𝑛! =6

Aftercompletingthetable,havestudentsmoveontowritingthegeneralrule.Remembertheexponentruleformultiplyingwiththesamebase:

𝒏𝒙 ∙ 𝒏𝒚 = 𝒏𝒙!𝒚

Writethematchingruleforlogarithms:

𝐥𝐨𝐠 𝒂 ∙ 𝒃 = 𝐥𝐨𝐠𝒂 + 𝐥𝐨𝐠𝒃

8)Explainwhateachoftheserulesmeansusingcompletesentences.Whenmultiplyingtwopowers,ifthebaseisthesame,thebaseremainsthesameandweaddtheexponents.Multiplicationinsidethelogcanbeturnedintoadditionoutsidethelog.

IMPActivity:ComparingExponentialandLogarithmicRules TeacherDirections8 !

Note:Youmaywanttoaskstudentsiftheoppositeistrue,ifweareaddingtwologarithmicexpressions,canwerewritethemastheproductofalog?(Yes)Infact,thereasonlogarithmswereinvented(Napier,https://en.wikipedia.org/wiki/History_of_logarithms#Tables_of_logarithms)wastomakemultiplicationeasier!]

9)Makeoneexampleofyourownforeachoftheselaws.Answersmayvary.Haveseveralstudentsbringuptheirexamplesandasktheclasstovoteastowhethertheexamplesaccuratelyrepresenteachoftherules.

Task3:Usingsomeotherexamplestodiscoverasecondloglaw…Again,youwillneedtohaveawholeclassdiscussionaboutthepatternandrule,anddecidewhenyourclassisreadyforthatdiscussion.Theexplanationsthatstudentswritecanbeusedasformativeassessmentbyeitherselectingstudentsatrandomtoreadwhattheyhavewrittenorbycollectingit.Expand4tothe3rdpower,thenusetherule

fromthepreviouspagetofindthelog.

log! 4!

log!(64)

6

Comparethistofindingthelogof4.log! 4 =2Howcanyouusethe3?Wecanusethe3andmultiplyitby2tofindthesolution,6.

Expand1,000tothe3rdpower,thenusethe

rulefromthepreviouspagetofindthelog.

log!" 1,000!

log!" 1,000,000,000

9

Comparethistofindingthelogof1,000log!" 1000 =3Howcanyouusethe3?Wecanusethe3andmultiplyitby3tofindthesolution,9.

Expandn4tothe5thpower,thenusetherule

fromthepreviouspagetofindthelog.

log! 𝑛! !

log! 𝑛!"

Comparethistofindingthelogof𝑛!.log! 𝑛! =4Howcanyouusethe5?Wecanusethe5andmultiplyitby4tofindthesolution,20.

IMPActivity:ComparingExponentialandLogarithmicRules TeacherDirections9 !

Remembertheexponentruleforraisingapowertoapower:

𝒏𝒙 𝒚 = 𝒏𝒙𝒚

Writethematchingruleforlogarithms:

𝐥𝐨𝐠 𝒂𝒃 = 𝒃 ∙ 𝒍𝒐𝒈 𝒂

10)Explainwhateachoftherulesatthebottomofthepreviouspagemeansusingcompletesentences.Whenapowerisraisedtoapower,thebaseremainsthesameandwemultiplythepowers.Anexponentinsidealogcanbemovedoutfrontasamultiplier.

11)Makeoneexampleofyourownforeachoftheselaws.Answersmayvary.Haveseveralstudentsbringuptheirexamplesandasktheclasstovoteastowhethertheexamplesaccuratelyrepresenteachoftherules.12)Usingwhatyouhavelearnedabouttheproductruleforlogs,iflog 𝑎 ∙ 𝑏 = log𝑎 + log 𝑏,thenhowmightwebeabletorewritelog !

!?

Studentsshouldbeabletoreasonthatmultiplicationbecameaddition,sodivisionwouldbesubtraction. log !

!= log𝑎 − log 𝑏

Youmightalsowantstudentstotryafewexamples.