Graphing Exponential and Logarithmic Functions

Objective I can graph exponential functions using a graphing utility and identify asymptotes, intercepts, domain and range.

Characteristics of Exponential Functions For exponential functions of form f(x) = b x Domain: {x | -∞ ≤ x ≤ ∞} all real numbers Range: {y | y > 0 } all positive real numbers Graph of f(x) = b x passes thru (0,1) [ f(0) = b 0 = 1, b ≠ 0 ] Y-intercept is (0,1) No X-intercept If b > 1, f(x) = b x goes up to right, increasing function. The greater the b value, the steeper the increase If 0 < b < 1, f(x) = b x goes down to right, decreasing function. The smaller the b value, the steeper the decrease f(x) = b x is one to one and has inverse function Graph of f(x) = b x Approaches but does not touch x axis. Horizontal asymptote : y=0

Characteristics of Logarithmic Functions For logarithmic functions of the form f(x) = log b (x) Domain: {x | x > 0} all real numbers (0, ∞) Range: {y | -∞ ≤ y ≤ ∞} all real numbers (-∞, ∞) Graph of f(x) = log b (x) passes thru (1,0) [ f(1) = log b (1) = 0] No Y-intercept X-intercept is (1,0) If b > 1, f(x) = log b (x) goes up to right, increasing function. If 0 < b < 1, f(x) = log b (x) goes down to right, decreasing function. f(x) = log b (x) is one to one and has an inverse function f(x) = log b (x) is the inverse function of f(x) = b x – If f(x) = b x, then f -1 (x) = log b (x) Graph of f(x) = log b (x) approaches but does not touch y axis. Vertical asymptote : x=0 (y axis)

Graphing Exponential Functions y = 3 x y = (1/3) x y = -3 x -1 y = (1/3) x +2

Graphing Logarithmic Functions

Graphing Exponential Functions

Graphing Logarithmic Functions

y = 3 x y = (1/3) x y = -3 x -1 y = (1/3) x +2

Graphing Exponential Function f(x) = (3) x XY Domain Range Asymptotes y x

Properties of Logarithms log a 1 = 0 because a 0 = 1. log a a = 1 because a 1 = a. log a a x = x and a log a x = x.Inverse Properties If log a x = log a y, then x = y.One-to-one Property

Properties of Logarithms Properties of Natural Logarithms In 1 = 0 because e 0 = 1 In e = 1 because e 1 = e ln e x = x and e ln x = xInverse Properties If In x = In y, then x = yOne-to-one Property

Properties of Logarithms Laws of Logarithms If M and N are positive real numbers and b is a positive number other than 1, then: log b MN = log b M + log b NLog of a Product log b M/N = log b M - log b NLog of a Quotient log b M = log b N if and only if M = N Identity (one-to-one) log b M k = k log b N, for any real number kLog of a Power

Properties of Logarithms Change of Base Property For any logarithmic bases a and b, and any positive number M, log b M = log a M ________________ log a b Introducing commom logarithms:log b M = log M log b Introducing natural logarithms:log b M = ln M ln b

Example 1 Use the properties to expand an expression: log 4 5x 3 y = Log of a Product Log of a Power

Example 2 Use the properties to condense an expression: (1/2) log x + 3 log (x+1) = Log of a Power Log of a Product

Example 3 Use the properties to solve a natural logarithm: e x = 20 Take natural log of each side Inverse Property Solve

Example 4 Use the properties to solve a natural logarithm: 4e 2x -3 = 2 Add Divide Take natural log of each side Inverse Property Divide Solve