Exponential functions in Economics: Economics applications of ln Interest compounding: for an interest rate r compounded Alternative definition of rate of growth: since the rate of community project at frequency m on an initial principal A, the value of the 1 df d ln (f(t)) r mt growth is = , it can also be expressed as asset at time t is V (m; t) = A 1 + . In the limit f(t) dt dt m d ln (f(t)) rt . mathcentre community project m ! 1 we have: V (t) = Ae . The rate r can take dt negative values in the case of deflation or depreciation. encouraging academics to share maths support resources Elasticity: the elasticity of a function y(x) with respect to All mccp resources are released under a Creative Commons licence Rate of growth: for a function f(t), the rate of growth is d ln y x dy 1 df x is = . If y is an exponential function of x, defined as . In the case f(t) represents an expo- d ln x y dx f(t) dt then the elasticity is the slope of the straight line obtained mcccp-richard-4 nential growth and takes the form Aert then the rate of when plotting y as a function of x on a log-log graph (which Arert is the same as plotting ln y as a function of ln x). For the help you need to support your course growth is = r. Aert Exponential and Logarithm for Economics and The Cobb-Douglas Production functions: are widely com- Business Studies 5 mon in Economics and are a family of functions taking the 4 exp(x) This leaflet is an overview of the properties of the functions form: Q = AKαLβ. and and their applications in Economics. 3 e ln Logarithm 2 Author: Morgiane Richard, University of Aberdeen The logarithm function log in base b is the inverse function 1 ln(x) Reviewers: Anthony Cronin (University College Dublin), of the exponential function in base b: 0 Shazia Ahmed (University of Glasgow) y y = logb x , x = b −1 Exponential −2 The natural logarithm, ln is the inverse function of the t −3 The exponential function is f(t) = b , where b > 1 is called natural exponential function: −4 the base. y y = ln x , x = e −5 The most commonly occurring base in Business and Eco- −1 0 1 2 3 4 5 nomics is e ≈ 2:72 and the corresponding exponential func- This means that eln x = ln ex = x and for any base b, t logb x x x tion is the natural exponential functionf(t) = e = exp (t). b = logb b = x. Figure 1: Graph of the functions e and ln x. m 1 Properties of ln: The number e is defined as e = lim 1 + . m!1 m ln (ut) = ln u + ln t Properties of e: 1 t ln ( ) = − ln t and ln ( ) = ln t − ln u (et)u = (eu)t = eut t u www.mathcentre.ac.uk u et+u = eteu ln (t ) = u ln t 1 et log t = log t log c and log t = ln t log e (conversion of e−t = and et−u = b c b b b et eu base) 0 d(et) d(ef(t)) d ln(t) 1 d ln (f(t)) f (t) 1 df(t) = et and = f 0(t)ef(t) = and = = dt dt dt t dt f(t) f(t) dt 1.