2. Copyright © 2007 Pearson Education, Inc. Slide 5-2 Chapter 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

3. Copyright © 2007 Pearson Education, Inc. Slide 5-3 5.4 Logarithmic Functions The function f (x) = a x, a  1, is one-to-one and thus has an inverse. The logarithmic function with base a and the exponential function with base a are inverse functions. So,

4. Copyright © 2007 Pearson Education, Inc. Slide 5-4 5.4 Graphs of Logarithmic Functions Recall that the graph of the inverse function is reflexive about the line y = x. The figure above is the typical shape for such graphs where a > 1 (includes base e and base 10 graphs).

5. Copyright © 2007 Pearson Education, Inc. Slide 5-5 5.4 Graphs of Logarithmic Functions Below are typical shapes for such graphs where 0 < a < 1.

6. Copyright © 2007 Pearson Education, Inc. Slide 5-6 5.4 The Logarithmic Function: f (x) = log a x, a > 1

7. Copyright © 2007 Pearson Education, Inc. Slide 5-7 5.4 The Logarithmic Function: f (x) = log a x, 0 < a < 1

8. Copyright © 2007 Pearson Education, Inc. Slide 5-8 5.4 Determining Domains of Logarithmic Functions ExampleFind the domain of each function. Solution (a)Argument of the logarithm must be positive. x – 1 > 0, or x > 1. The domain is (1,  ). (b)Use the sign graph to solve x 2 – 4 > 0. The domain is (– ,–2) (2,  ).

9. Copyright © 2007 Pearson Education, Inc. Slide 5-9 5.4 Graphing Translated Logarithmic Functions ExampleGive the domain, range, asymptote, and x-intercept. (a) Solution (a)The argument x – 1 shifts the graph of y = log 2 x 1 unit to the right. –Vertical asymptote: x = 1 –x-intercept: (2,0) –Domain: (1,  ), Range: (– ,  )

10. Copyright © 2007 Pearson Education, Inc. Slide 5-10 5.4 Graphing Translated Logarithmic Functions (b)Here, 1 is subtracted from y = log 3 x shifting it down 1 unit. –Vertical asymptote: y-axis (or x = 0) –x-intercept : (3,0) –Domain: (0,  ), Range: (– ,  )

11. Copyright © 2007 Pearson Education, Inc. Slide 5-11 5.4 Determining Symmetry ExampleShow analytically that the graph of is symmetric with respect to the y-axis. Solution Since f (x) = f (–x), the graph is symmetric with respect to the y-axis.

12. Copyright © 2007 Pearson Education, Inc. Slide 5-12 5.4 Finding the Inverse of an Exponential Function ExampleFind the inverse function of Solution Replace f (x) with y. Interchange x and y. Isolate the exponential. Write in logarithmic form. Replace y with f –1 (x).

13. Copyright © 2007 Pearson Education, Inc. Slide 5-13 5.4 Logarithmic Model: Modeling Drug Concentration ExampleThe concentration of a drug injected into the bloodstream decreases with time. The intervals of time in hours when the drug should be administered are given by where k is a constant determined by the drug in use, C 2 is the concentration at which the drug is harmful, and C 1 is the concentration below which the drug is ineffective. Thus, if T = 4, the drug should be administered every 4 hours. For a certain drug, k = C 2 = 5, and C 1 = 2. How often should the drug be administered?

14. Copyright © 2007 Pearson Education, Inc. Slide 5-14 5.4 Logarithmic Model: Modeling Drug Concentration SolutionSubstitute the given values into the equation. The drug should be given about every 3 hours.