BEE1024 Mathematics For Economists

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ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesBEE1024 Mathematics for EconomistsExponential and logarithmic functions, ElasticitiesJuliette Stephenson and Amr (Miro) AlgarhiAuthor: Dieter BalkenborgDepartment of Economics, University of ExeterWeek 5BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticities1Objectives2The Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function3Logarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functions4ApplicationsCompounded InterestsExponential decayThe logistic curveBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic tial functions: describes growth processes withconstant growth ratepopulation growth, growth of GDP, in‡ation etc.BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic tial functions: describes growth processes withconstant growth ratepopulation growth, growth of GDP, in‡ation etc.logarithm: the exponent required to produce a given numberinverse function, transforms multiplication into addition:10a 10b 10a bLogarithmic di erentiationBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic tial functions: describes growth processes withconstant growth ratepopulation growth, growth of GDP, in‡ation etc.logarithm: the exponent required to produce a given numberinverse function, transforms multiplication into addition:10a 10b 10a bLogarithmic di erentiationElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionThe Exponential Functionpower: x yBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionThe Exponential Functionpower: x ybase: xBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionThe Exponential Functionpower: x ybase: xindex or exponent: yBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionThe Exponential Functionpower: x ybase: xindex or exponent: ypower function like x 2 : vary xBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionThe Exponential Functionpower: x ybase: xindex or exponent: ypower function like x 2 : vary xexponential function 2y : vary yBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionThe Exponential Functionpower: x ybase: xindex or exponent: ypower function like x 2 : vary xexponential function 2y : vary yadmissible values for y : positive integers, integers, rationals,real numbersBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionThe Exponential Functionpower: x ybase: xindex or exponent: ypower function like x 2 : vary xexponential function 2y : vary yadmissible values for y : positive integers, integers, rationals,real numbersproblem: for general y the power x y can only be de ned forpositive xBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionPower functions424222018161412108642-4-20321-42 xy 2: f (x ) 4x2Balkenborg-202 xy 2:f (x ) x 2 41x2Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionPower 2x3412y 12 : f (x ) x 5pxBalkenborgy f (x ) x324532: x1pxExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionExponential Functionsapproximate irrational index y by fractionmn:mx y : mlim x n .n-2-1!y8866442201yx 3: g (y ) 23yBalkenborg-2x -113:0g (y ) y11 y32 3yExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionExponential Functions43211y0.80.6z xyBalkenborg1.5x0.5x22.530Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionProperties of exponential functionsAn exponential function ax :Calculational rules for generalized powers:as t as atast (as )tbut( as ) t 6 a (sBalkenborg(ab )s as bst)Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionProperties of exponential functionsAn exponential function ax :is strictly convex and has strictly positive values;Calculational rules for generalized powers:as t as atast (as )tbut( as ) t 6 a (sBalkenborg(ab )s as bst)Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionProperties of exponential functionsAn exponential function ax :is strictly convex and has strictly positive values;is for a 1 strictly increasing with limx !limx ! ax ; ax 0 andCalculational rules for generalized powers:as t as atast (as )tbut( as ) t 6 a (sBalkenborg(ab )s as bst)Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionProperties of exponential functionsAn exponential function ax :is strictly convex and has strictly positive values;is for a 1 strictly increasing with limx !limx ! ax ;is for 0 a 1: decreasing with limx !limx ! ax 0. ax 0 andax andCalculational rules for generalized powers:as t as atast (as )tbut( as ) t 6 a (sBalkenborg(ab )s as bst)Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number ePut P0 0 (the principal) in savings accountBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number ePut P0 0 (the principal) in savings account xed nominal annual interests rate r 0BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number ePut P0 0 (the principal) in savings account xed nominal annual interests rate r 0Interests paid n times during the yearBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number ePut P0 0 (the principal) in savings account xed nominal annual interests rate r 0Interests paid n times during the yearamount Pt in your savings account after t years:BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number ePut P0 0 (the principal) in savings account xed nominal annual interests rate r 0Interests paid n times during the yearamount Pt in your savings account after t years:formula for compounded interestsPt P0 1 Balkenborgr ntnExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number ePut P0 0 (the principal) in savings account xed nominal annual interests rate r 0Interests paid n times during the yearamount Pt in your savings account after t years:formula for compounded interestsPt P0 1 rnr ntninterest paid per periodBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number ePut P0 0 (the principal) in savings account xed nominal annual interests rate r 0Interests paid n times during the yearamount Pt in your savings account after t years:formula for compounded interestsPt P0 1 rnr ntninterest paid per periodnt total number of interest payments.BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number eThe (natural) exponential function:balance in account after one year if interests paid continuously:exp (r ) limn ! 1 The table below shows the value of 1 nnnnnnnr 5.4% 41.055103375 121.055356752 3641.055480375 87361.055484426 524 1601.055484599 31 449 600 1.055484602! 1.055484602Balkenborgr nnr nn.for various n and r :r 565406121.0565406131.056540615r 82792352.7182817962.718281828Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number eThe Euler number e is de ned ase exp (1) limn ! 1 1 nn.The ‘natural exponential function’is indeed the exponentialfunction with base e:exp (r ) e r“proof” for rational r :rr n lim1 m ! ,n rmn ! nn1 rm1 lim 1 lim1 m ! m ! mmm r1 er lim 1 m ! mexp (r ) lim1 Balkenborgnm rExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionCompounded interests and the number eformula for continuously compounded interests:Pt P0 e rt .BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionProperties of the natural exponential function123e 0 1,e1 ee x 0 for all x4d (e x )dx exIn particular, e x is strictly increasing and convex.instantaneous growth rate of a function y f (x ): dyydxwhen x is increased by a exponential function has constantgrowth rate 1.5(e a )b e ab .e a b e a e bIn particular1ex ex(because eBalkenborgx ex e 0 1).Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionnLet fn (x ) 1 xn .Intuition for property 3: 1 xn is positive when x 0 or when nlarge compared to jx j . Then fn (x ) 0 and so hence e x 0.Intuition for property 4:xdfn n 1 dxnn 11x 1 nnn 1for n very large compared to jx j since 1 Balkenborg1 xnxnn fn ( x )is then very close to 1.Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionTheoremThere is one and only one function y f (x ) which satis es the“initial condition” f (0) 1 and the “di erential equation”dy ydxand this is the exponential function f (x ) exp x e x .BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionProperties of the natural exponential functionExponential versus polynomial growth: For any polynomialP (x )ex limx ! P (x )BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionIntuition: Suppose P (x ) am x m . . . has degree m.nApproximate e x by 1 xn with n larger than m. Thenexlimx ! P (x )lim Cx n mx ! limx ! 1 xnP (x ) Balkenborgn limx ! 1 nnxn . . . am x m . . .Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionA quicker way to calculate e x : is to use the formulaex 1 x x2x3xn . .2!3!n!BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsLogarithmic functionsLet a 0. The logarithmic function loga x to the base a is de nedas the inverse of the exponential function ayy loga x , ay xFor instance, 1000 103 , so log10 1000 3;log2 81 3.Balkenborg18 23,soExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsThe natural logarithmnatural logarithm functiony ln (x ) , x e y432-3-2-101x234-1-2-3BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsDi erentiating the natural logarithmx e ln x(ln x )0 1e ln x 1 xx1because by the chain rule1 dx e ln xdxBalkenborg(ln x )0 x(ln x )0Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsProperties of the natural logarithm1ln (y ) is only de ned for strictly positive y 0.BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsProperties of the natural logarithm1ln (y ) is only de ned for strictly positive y 0.2d ln (y )dy y1 . In particular, ln (y ) is strictly increasing andconcave.BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsProperties of the natural logarithm1ln (y ) is only de ned for strictly positive y 0.2d ln (y )dy3 y1 . In particular, ln (y ) is strictly increasing andconcave.ln (1) 0, ln (e ) 1.BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsProperties of the natural logarithm1ln (y ) is only de ned for strictly positive y 0.2d ln (y )dy34 y1 . In particular, ln (y ) is strictly increasing andconcave.ln (1) 0, ln (e ) 1.limy !0 ln (y ) , limy ! ln (y ) .BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsProperties of the natural logarithm1ln (y ) is only de ned for strictly positive y 0.2d ln (y )dy345 y1 . In particular, ln (y ) is strictly increasing andconcave.ln (1) 0, ln (e ) 1.limy !0 ln (y ) , limy ! ln (y ) .ln (ab ) ln (a) ln (b ), ln ab b ln (a). In particularln 1a ln (a).BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsProperties of the natural logarithm1ln (y ) is only de ned for strictly positive y 0.2d ln (y )dy3456 y1 . In particular, ln (y ) is strictly increasing andconcave.ln (1) 0, ln (e ) 1.limy !0 ln (y ) , limy ! ln (y ) .ln (ab ) ln (a) ln (b ), ln ab b ln (a). In particularln 1a ln (a).Logarithmic differentiation: Combined with the chainrule one obtains the following useful formula where y g (x )is any di erentiable function:d ln (g (x ))dxBalkenborg g 0 (x )g (x )(1)Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsTheoremThe natural logarithm y (x ) ln x is the unique solution to thedi erential equation1dy dxxwhich satis es the initial condition y (1) 0.BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsDi erentiating general exponential and logarithmicfunctionsln (x y ) y ln (x )general powers can be rewritten asx y e y ln (x ) e indexln (base ) .Partial di erentiation yields x y y x y x e y ln (x ) ln (x ) x y ln (x ) e y ln (x ) yBalkenborg1x yx y1 yx yx1.Exponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsDerivative of an exponential function y ax isda xdx ln (a) ax .The instantaneous growth rate of y ax is ln (a)The derivative of a power function y x b isdx bdx bx b1even if b is irrational.BalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsx loga (y ) , y ax .We havey ax , y e x ln (a ) , ln (y ) x ln (a) , x ln (y )ln (a)sologa y ln (y )ln (a )loga y has hence the derivatived (log a (y ))dyBalkenborg 1ln (a )yExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesCompounded InterestsExponential decayThe logistic curveGliederung12345ObjectivesThe Exponential FunctionDe nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential functionLogarithmic functionsDe nitionDi erentiating the natural logarithmProperties of the natural logarithmDi erentiating general exponential and logarithmic functionsApplicationsCompounded InterestsExponential decayThe logistic curveElasticitiesOwn-price elasticityElasticities and logarithmsOther ElasticitiesBalkenborgExponential and logarithmic functions, Elasticities

ObjectivesThe Exponential FunctionLogarithmic functionsApplicationsElasticitiesCompounded InterestsExponential decayThe logistic curveCompounded InterestsExample: Suppose Bank A o ers the annual nominal

The Exponential Function Logarithmic functions Applications Elasticities BEE1024 Mathematics for Economists Exponential and logarithmic functions, Elasticities Juliette Stephenson and Amr (Miro) Algarhi Author: Dieter Balkenborg Department of Economics, University of Exeter Week 5 Balkenborg Exponential and logarithmic functions, Elasticities

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