07 Comparing Exponential and Logarithmic Rules · IMP Activity: Comparing Exponential and...
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IMP Activity: Comparing Exponential and Logarithmic Rules 1 Comparing Exponential and Logarithmic Rules Task 1: Looking closely at exponential and logarithmic patterns… 1) In a prior lesson you graphed and then compared an exponential function with a logarithmic function and found that the functions are __________________________ functions. 2) When a function is the inverse of another function we know that if the _______________ of one function maps onto the output of another function then the inverse maps the _________________ to the ________________. 3) Given the function f(x)=2 x and a table with corresponding values x -2 -1 0 1 2 3 f(x) 1 4 1 2 1 2 4 8 complete the table for the inverse function, !! () = ! . x !! () 4) Using the tables, if 2 x = 8, then x = ______________. If ! = 3, then y = ______________. 5) Describe what you noticed about the relationship between the exponential and logarithmic equations in number 4. _____________________________________________________________________________ ______________________________________________________________________________ 6) Once more, if 2 x = 2, then x = ______________. If ! = 1, then y = ______________. 7) Simplify the following: a) ! 4 = ______ b) ! 27 = ______ c) ! 16 = ______ Name _____________________________________ Date _____________ Period ___________ Let’s make a rule for this relationship….. If = , then ______ = _______. We read _____ = ______ as __________________________________________________.
Transcript of 07 Comparing Exponential and Logarithmic Rules · IMP Activity: Comparing Exponential and...
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.
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