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Lecture 4. Exponential and Logarithmic Functions. 1 / 11 Exponential Functions Denition (Exponential Functions) If b > 0 and b 6= 1, then there is a unique function called exponential function with base b that is dened by f (x ) = b x for every real number x . Theorem (Properties of an Exponential Function) The exponential function f (x ) = bx for b > 0 and b 6= 1 has these properties: 1 It is dened, continuous, and positive f (x ) > 0 for all x ∈ R. 2 The x axis is a horizontal asymptote of the graph of f . 3 The y intercept of the graph is (0, 1); there is no x intercept. 4 If b > 1, lim bx = 0 and lim bx = +∞, x →−∞ If 0 < b < 1, lim bx = +∞ and x →−∞ 5 x →+∞ lim bx = 0. x →+∞ For all x, the function is increasing (graph rasing) if b > 1 and decreasing (graph falling) if 0 < b < 1. 2 / 11 Exponential Functions Denition (Euler's Number) e := 1 + 1 1 1 + + + ... 1 1·2 1·2·3 ≈ 2.71828182845904523536028747135266249775724709369995 3 / 11 Exponential Functions y 1 x 3 1 x e 1 x 2 −3 −2 −1 9 8 7 6 5 4 3 2 1 0 3x ex 2x 1 2 3 x Graphs of exponential functions 4 / 11 Logarithmic Functions Denition (Exponential Functions) If b > 0, then the logarithm of x to the base b (b > 0, b 6= 1), denoted logb x , is the number y such that by = x ; that is, y = logb x if and only if by = x for x > 0. Example log10 1, 000 = 3 since 103 = 1, 000. log2 32 = 5 since 25 = 32. log10 1 125 = −3 since 5−3 = 1 125 . 5 / 11 Logarithmic Functions Theorem (Logarithmic Rules) Let b be any logarithmic base (b > 0, b 6= 1). Then logb 1 = 0 and logb b = 1 and if u and v are any positive numbers, we have The equality rule logb u = logb v if and only if u = v The product rule logb (uv ) = logb u + logb v The power rule logb u r = r · logb u for any real number r The quotient rule logb The inversion rule logb b = u u v u = logb u − logb v 6 / 11 Logarithmic Functions Theorem (Properties of an Logarithmic Function) The logarithmic function f (x ) = logb x for b > 0 and b 6= 1 has these properties: 1 It is dened, continuous for all x ∈ R. 2 The y axis is a vertical asymptote of the graph of f . 3 The x intercept of the graph is (1, 0); there is no y intercept. 4 If b > 1, lim+ logb x = +∞ and lim logb x = +∞, x →0 x →+∞ If 0 < b < 1, lim bx = +∞ and x →0+ 5 lim logb x = +∞. x →+∞ For all x > 0, the function is increasing (graph rasing) if b > 1 and decreasing (graph falling) if 0 < b < 1. 7 / 11 Exponential Functions Denition (Natural Logarithm) The logarithm loge x is called the natural logarithm of x and is denoted by ln x that is, for x > 0 y = ln x if and only if e y = x Theorem (Conversion Formula for Logarithms) If a and b are positive numbers with b 6= 1, then logb a = ln a . ln b 8 / 11 Logarithmic Functions y log2 x 3 lnx log3 x 2 1 0 1 2 3 4 5 6 7 8 9 x −1 −2 −3 log 13 x log e1 x log 12 x Graphs of logarithmic functions 9 / 11 Dierentiation of Exponential and Logarithmic Functions Theorem (The Derivative of e x ) For every real number x (e x )0 = e x Theorem (The Chain Rule for e u ) If u (x ) is a dierentiable function of x, then e u (x ) 0 = e u(x ) · u 0 (x ). Theorem (The Derivative of ln x ) For all x > 0 (ln x )0 = 1 x Theorem (The Chain Rule for ln u ) If u (x ) is a dierentiable function of x, then (ln u (x ))0 = u 0 (x ) for u (x ) > 0. u (x ) 10 / 11 Thank you for your attention! 11 / 11