2. Copyright © 2011 Pearson Education, Inc. Logarithmic Functions and Their Applications Section 4.2 Exponential and Logarithmic Functions

3. 4.2 Copyright © 2011 Pearson Education, Inc. Slide 4-3 Since exponential functions are one-to-one functions, they are invertible. The inverses of the exponential functions are called logarithmic functions. If f(x) = a x, then instead of f –1 (x), we write log a (x) for the inverse of the base-a exponential function. We read log a (x) as “log of x base a,” and we call the expression log a (x) a logarithm. In general, log a (x) is the exponent that is used on the base a to obtain the value x. Since the exponential function f (x) = a x has domain (– ∞, ∞) and range (0, ∞), the logarithmic function f(x) = log a (x) has domain (0, ∞) and range (– ∞, ∞). So, there are no logarithms of negative numbers or zero. The Definition

4. 4.2 Copyright © 2011 Pearson Education, Inc. Slide 4-4 Definition: Logarithmic Function For a > 0 and a ≠ 1, the logarithmic function with base a is denoted as f(x) = log a (x), where y = log a (x) if and only if a y = x. Note that log a (1) = 0 for any base a, because a 0 = 1 for any base a. There are two bases that are used more frequently than others; they are 10 and e. The notation log 10 (x) is abbreviated log(x) and log e (x) is abbreviated ln(x). These are called the common logarithmic function and the natural logarithmic function, respectively. The Definition

5. 4.2 Copyright © 2011 Pearson Education, Inc. Slide 4-5 The functions of y = a x and y = log a (x) for a > 0 and a ≠ 1 are inverse functions. So, the graph of y = log a (x) is a reflection about the line y = x of the graph of y = a x. The graph of y = a x has the x-axis as its horizontal asymptote, while the graph of y = log a (x) has the y-axis as its vertical asymptote. Graphs of Logarithmic Funcitons

6. 4.2 Copyright © 2011 Pearson Education, Inc. Slide 4-6 Properties of Logarithmic Functions The logarithmic function f(x) = log a (x) has the following properties: 1. The function f is increasing for a > 1 and decreasing for 0 < a < 1. 2. The x-intercept of the graph of f is (1, 0). 3. The graph has the y-axis as a vertical asymptote. 4. The domain of f is (0, ∞), and the range of f is (– ∞, ∞). 5. The function f is one-to-one. 6. The functions f(x) = log a (x) and f(x) = a x are inverse functions. Graphs of Logarithmic Funcitons

7. 4.2 Copyright © 2011 Pearson Education, Inc. Slide 4-7 Any function of the form g(x) = b · log a (x – h) + k is a member of the logarithmic family of functions. The graph of f moves to the left if h 0. The graph of f moves upward if k > 0 or downward if k < 0. The graph of f is stretched if b > 1 and shrunk if 0 < b < 1. The graph of f is reflected in the x-axis if b is negative. The Logarithmic Family of Functions

8. 4.2 Copyright © 2011 Pearson Education, Inc. Slide 4-8 One-to-One Property of Logarithms For a > 0 and a ≠ 1, if log a (x 1 ) = log a (x 2 ), then x 1 = x 2. Logarithmic and Exponential Equations