Logarithms and Exponential Functions. Gerda de Vries & John S. Macnab. match as necessary, or to work these results into other lessons.
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1 Logritms nd Eponentil Functions Gerd de Vries & Jon S. Mcn It is epected tt students re lred fmilir wit tis mteril. We include it ere for completeness. Te tree lessons given ere re ver sort. Te tecer is encourged to mi nd mtc s necessr, or to work tese results into oter lessons. Lesson : (Review nd introducing e For tis module, students will use nturl logritms. Nottion: ln(=log e ( Recll tt If f(=, for >, ten f - ( = log ( Alterntivel, te ide is sometimes given s conversion of sttements from eponentil form to logritmic form nd vice vers. Eponentil Form Logritmic Form log 2 (32 5 log ( log( 2 c log ( c Prt Log nd Eponentil Functions.doc Pge of 8
2 Te rules of eponents trnslte to te rules of logritms. Eponentil Rule Logritmic Rule log ( log ( log ( log log ( log ( log log ( Cnge of Bse Te cnge of se formul is simplicit itself. log ( Proof: log ( log ( log (, if > so, log ( log ( simplifing: log ( log ( nd log (, wic is te desired result. log ( Introducing e Consider popultion tt doules in er. An investment tt doules is nice tougt s well. We will just look t one er. If N is te numer of orgnisms in te popultion, ten N N ( N 2N Prt Log nd Eponentil Functions.doc Pge 2 of 8
3 Or, to put it noter w, if we consider te growt using te compound interest formul, wen te rte of growt is % nd te growt is compounded nnull, te popultion doules in one er. Tis is not surprising. Now, suppose we cnge te prolem, so tt te popultion increse is compounded twice in te er. N N 2 2 ( N 2.25N Wen te growt is compounded twice nnull, te increse is of 5%, ut it ppens twice. Insted of douling, te popultion increses 25%. Question: Wt ppens s te compounding period gets smller nd te numer of times te growt is compounded increses? Period p (ers Formul Popultion Size N 3 N 2.37N N N ( 3 4 N 2.44N N N ( 4 5 N 2.4N N N ( 5 N 2.59N N N ( Does te popultion grow witout ound, or is tere mimum growt tt is possile under tese conditions? Prt Log nd Eponentil Functions.doc Pge 3 of 8
4 To put te question noter w, does te it p N eist? If so, wt is its vlue? It is not simple to get n ect p p vlue, ut students sould e le to grp te function nd mke te inference tt te grp ppers to pproc te orizontl smptote We use te letter e to represent tis numer. e p e p p Prt Log nd Eponentil Functions.doc Pge 4 of 8
5 Lesson 2: Derivtives of Eponentil Functions Let f ( e. Consider te grp of =e Oservtions: is lws positive is lws incresing d s increses, increses ever more rpidl d d s pproces, pproces zero d Prt Log nd Eponentil Functions.doc Pge 5 of 8
6 From te definition of te derivtive: d d d d d d d d e e ( e e f ( e ( e f ( So, if d d e ten e (some numer, given ( e. Recll our sic definition: e p p p Let, so tt p e Rising ot sides to te power of : e Prt Log nd Eponentil Functions.doc Pge 6 of 8
7 So now ck to te derivtive: d( e e d e e e ( e (( It sould e now cler w =e is suc useful function: it is its own derivtive. We cn now tke te derivtive of n eponentil function wit positive se, simpl converting it to function wit se e nd using te cin rule. e.g. e d d d d ln( ln( ln( e ln( ln( ln( In generl, if >, ten d d ln( Prt Log nd Eponentil Functions.doc Pge 7 of 8
8 Lesson 3: Derivtives of Logritmic Functions Tese re perps esiest to introduce troug implicit differentition. ln( e d( e d( d d d e d d d e d d As efore, we cn cnge se to see tt for n se >, ln( d dlog ( ln( d d d(ln( ln( d ln( ln( Note tt ln( log ( e, so we could lso write d(log ( log d ( e Prt Log nd Eponentil Functions.doc Pge 8 of 8
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