Exponential and logarithm functions mcTYexplogfns Exponentialfunctionsandlogarithmfunctionsareimporta ntinboththeoryandpractice
120K - views

Exponential and logarithm functions mcTYexplogfns Exponentialfunctionsandlogarithmfunctionsareimporta ntinboththeoryandpractice

Inthis unitwelookatthegraphsofexponentialandlogarithmfunct ionsandseehowtheyarerelated Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideot

Download Pdf

Exponential and logarithm functions mcTYexplogfns Exponentialfunctionsandlogarithmfunctionsareimporta ntinboththeoryandpractice




Download Pdf - The PPT/PDF document "Exponential and logarithm functions mcTY..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.



Presentation on theme: "Exponential and logarithm functions mcTYexplogfns Exponentialfunctionsandlogarithmfunctionsareimporta ntinboththeoryandpractice"— Presentation transcript:


Page 1
Exponential and logarithm functions mc-TY-explogfns-2009-1 Exponentialfunctionsandlogarithmfunctionsareimporta ntinboththeoryandpractice.Inthis unitwelookatthegraphsofexponentialandlogarithmfunct ions,andseehowtheyarerelated. Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature. Afterreadingthistext,and/orviewingthevideotutorialo nthistopic,youshouldbeableto: specifyforwhichvaluesof theexponentialfunction ) = maybedefined, recognizethedomainandrangeofanexponentialfunction,

identifyaparticularpointwhichisonthegraphofeveryexp onentialfunction, specifyforwhichvaluesof thelogarithmfunction ) = log maybedefined, recognizethedomainandrangeofalogarithmfunction, identifyaparticularpointwhichisonthegraphofeverylog arithmfunction, understandtherelationshipbetweentheexponentialfunct ion ) = andthenatural logarithmfunction ) = ln Contents 1. Exponentialfunctions 2. Logarithmfunctions 3. Therelationshipbetweenexponentialfunctionsand logarithmfunctions www.mathcentre.ac.uk 1 math centre2009
Page 2
1. Exponential functions Considerafunctionoftheform ) = ,where

a > .Suchafunctioniscalledan exponential function.Wecantakethreedifferentcases,where = 1 < a < and a > If = 1 then ) = 1 = 1 Sothisjustgivesustheconstantfunction ) = 1 Whathappensif a > ?Toexaminethiscase,takeanumericalexample.Supposetha = 2 ) = 2 (0) = 2 = 1 (1) = 2 = 2 1) = 2 = 1 (2) = 2 = 4 2) = 2 = 1 (3) = 2 = 8 3) = 2 = 1 Wecanputtheseresultsintoatable,andplotagraphofthefu nction. ) = 2 Thisexampledemonstratesthegeneralshapeforgraphsoffu nctionsoftheform ) = when a > Whatistheeffectofvarying ?Wecanseethisbylookingatsketchesofafewgraphsofsimil ar functions. ) = 2 ) = 5 ) = 10

www.mathcentre.ac.uk 2 math centre2009
Page 3
Theimportantpropertiesofthegraphsofthesetypesoffunc tionsare: (0) = 1 forallvaluesof .Thisisbecause = 1 foranyvalueof forallvaluesof .Thisisbecause a > implies Whathappensif < a < ? Toexaminethiscase,takeanothernumericalexample. Supp ose that ) = (0) = = 1 (1) = 1) = = 2 (2) = 2) = = 4 (3) = 3) = = 8 Wecanputtheseresultsintoatable,andplotagraphofthefu nction. ) = Thisexampledemonstratesthegeneralshapeforgraphsoffu nctionsoftheform ) = when < a < Whatistheeffectofvarying ? Againwecanseebylookingatsketchesofafewgraphsof

similarfunctions. ) = ) = ( 10 ) = www.mathcentre.ac.uk 3 math centre2009
Page 4
Theimportantpropertiesofthegraphsofthesetypesoffunc tionsare: (0) = 1 forallvaluesof .Thisisbecause = 1 foranyvalueof forallvaluesof .Thisisbecause a > implies Noticethatthesepropertiesarethesameaswhen a > .Oneinterestingthingthatyoumight havespottedisthat ) = ( = 2 isareflectionof ) = 2 inthe axis,andthat ) = ( = 5 isareflectionof ) = 5 inthe axis. ) = ) = ) = 2 ) = 5 Ingeneral, ) = (1 /a isareflectionof ) = inthe axis. Aparticularlyimportantexampleofanexponentialfunctio nariseswhen

e.Youmightrecall thatthenumbereisapproximatelyequalto2.718.Thefuncti on ) = isoftencalled‘the exponentialfunction. Sincee and ,wecansketchthegraphsoftheexponential functions ) = and ) = = (1 ) = e ) = e www.mathcentre.ac.uk 4 math centre2009
Page 5
Key Point Afunctionoftheform ) = (where a > )iscalledanexponentialfunction. Thefunction ) = 1 isjusttheconstantfunction ) = 1 Thefunction ) = for a > hasagraphwhichisclosetothe -axisfornegative and increasesrapidlyforpositive Thefunction ) = for < a < hasagraphwhichisclosetothe -axisforpositive andincreasesrapidlyfordecreasingnegative

Foranyvalueof ,thegraphalwayspassesthroughthepoint (0 1) . Thegraphof ) = (1 /a isareflection,intheverticalaxis,ofthegraphof ) = Aparticularlyimportantexponentalfunctionis ) = ,wheree = 2 718 . . . . Thisisoften called‘the’exponentialfunction. 2. Logarithm functions Weshallnowlookatlogarithmfunctions.Thesearefunction softheform ) = log where a > .Wedonotconsiderthecase = 1 ,asthiswillnotgiveusavalidfunction. Whathappensif a > ?Toexaminethiscase,takeanumericalexample. Supposetha = 2 Then ) = log means x. Animportantpointtonotehereisthat,regardlessofthearg ument, . Soweshall

consideronlypositivearguments. (1) = log means (1) = 1 so (1) = 0 (2) = log means (2) = 2 so (2) = 1 (4) = log means (4) = 4 so (4) = 2 ) = log means = 2 so ) = ) = log means = 2 so ) = Wecanputtheseresultsintoatable,andplotagraphofthefu nction. www.mathcentre.ac.uk 5 math centre2009
Page 6
) = log Thisexampledemonstratesthegeneralshapeforgraphsoffu nctionsoftheform ) = log when a > Whatistheeffectofvarying ? Wecanseebylookingatsketchesofafewgraphsofsimilar functions.Forthespecialcasewhere e,weoftenwrite ln insteadof log ) = log ) = log ) = log = ln

Theimportantpropertiesofthegraphsofthesetypesoffunc tionsare: (1) = 0 forallvaluesof wemusthave x > forallvaluesof Whathappensif < a < ? Toexaminethiscase,takeanothernumericalexample. Supp ose that .Then ) = log means x. Animportantpointtonotehereisthat,regardlessofthearg ument, . Soweshall consideronlypositivearguments. www.mathcentre.ac.uk 6 math centre2009
Page 7
) = (1) = log means (1) = 1 so (1) = 0 (2) = log means (2) = 2 = so (2) = (4) = log means (4) = 4 = so (4) = ) = log means so ) = 1 ) = log means so ) = 2 Wecanputtheseresultsintoatable,andplotagraphofthefu nction. ) =

log 1/2 Thisexampledemonstratesthegeneralshapeforgraphsoffu nctionsoftheform ) = log when < a < Whatistheeffectofvarying ? Againwecanseebylookingatsketchesofafewgraphsof similarfunctions. ) = log 1/2 ) = log 1/5 ) = log 1/e www.mathcentre.ac.uk 7 math centre2009
Page 8
Theimportantpropertiesofthegraphsofthesetypesoffunc tionsare: (1) = 0 forallvaluesof wemusthave x > forallvaluesof Aninterestingthingthatyoumightwellhavespottedisthat ) = log isareflectionof ) = log inthe -axis and ) = log isareflectionof ) = log inthe -axis. ) = log 1/2 ) = log 1/5 ) = log ) = log

Generally, ) = log /a isareflectionof ) = log inthe -axis. Key Point Afunctionoftheform ) = log (where a > and = 1 )iscalledalogarithmfunction. Thefunction ) = log for a > hasagraphwhichisclosetothenegative -axisfor x < andincreasesslowlyforpositive Thefunction ) = log for < a < hasagraphwhichisclosetothepositive -axis for x < anddecreasesslowlyforpositive Foranyvalueof ,thegraphalwayspassesthroughthepoint (1 0) .Thegraphof ) = log /a isareflection,inthehorizontalaxis,ofthegraphof ) = log Aparticularlyimportantlogarithmfunctionis ) = log ,wheree = 2 718 . . . .Thisisoften

calledthenaturallogarithmfunction,andwritten ) = ln www.mathcentre.ac.uk 8 math centre2009
Page 9
3. The relationship between exponential functions and log- arithm functions Wecanseetherelationship betweentheexponentialfunctio ) = andthelogarithm function ) = ln bylookingattheirgraphs. ) = ln ) = e ) = Youcanseestraightawaythatthelogarithmfunctionisare ectionoftheexponentialfunction inthelinerepresentedby ) = . Inotherwords,theaxeshavebeenswapped: becomes ,and becomes Key Point Theexponentialfunction ) = istheinverseofthelogarithmfunction ) = ln Exercises

1.Sketchthegraphofthefunction ) = forthefollowingvaluesof ,onthesameaxes. (a) = 3 (b) = 6 (c) = 1 (d) (e) 2. Sketchthegraphofthefunction ) = log forthefollowingvaluesof ,onthesame axes. (a) = 3 (b) = 6 (c) (d) www.mathcentre.ac.uk 9 math centre2009
Page 10
3.Foreachofthefollowingpairsoffunctions,statewhethe rthegraphsarerelatedbyareflection inthe -axis,areflectioninthe -axis,areflectionintheline ) = ,areflectioninthe line ) = ,orthatthegraphsarenotrelatedbyanyofthesereflections (a) ) = 3 and ) = (b) ) = log and ) = 6 (c) ) = log and ) = (d) ) = log and )

= log (e) ) = and ) = Answers 1. ) = ) = ) = 3 ) = 6 ) = 1 2. ) = log 1/3 ) = log 1/6 ) = log ) = log www.mathcentre.ac.uk 10 math centre2009
Page 11
3. (a) Reflectinthe -axis (b) Reflectintheline ) = (c) Notrelatedbyanyofthesereflections (d) Reflectinthe -axis (e) Notrelatedbyanyofthesereflections www.mathcentre.ac.uk 11 math centre2009