1. Copyright © 2011 Pearson Education, Inc. Slide 5.4-1 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,

2. Copyright © 2011 Pearson Education, Inc. Slide 5.4-2 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).

3. Copyright © 2011 Pearson Education, Inc. Slide 5.4-3 5.4 Graphs of Logarithmic Functions Below are typical shapes for such graphs where 0 < a < 1.

4. Copyright © 2011 Pearson Education, Inc. Slide 5.4-4 5.4 The Logarithmic Function: f (x) = log a x, a > 1

5. Copyright © 2011 Pearson Education, Inc. Slide 5.4-5 5.4 The Logarithmic Function: f (x) = log a x, 0 < a < 1

6. Copyright © 2011 Pearson Education, Inc. Slide 5.4-6 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,  ).

7. Copyright © 2011 Pearson Education, Inc. Slide 5.4-7 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: (– ,  )

8. Copyright © 2011 Pearson Education, Inc. Slide 5.4-8 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: (– ,  )

9. Copyright © 2011 Pearson Education, Inc. Slide 5.4-9 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.

10. Copyright © 2011 Pearson Education, Inc. Slide 5.4-10 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).

11. Copyright © 2011 Pearson Education, Inc. Slide 5.4-11 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?

12. Copyright © 2011 Pearson Education, Inc. Slide 5.4-12 5.4 Logarithmic Model: Modeling Drug Concentration SolutionSubstitute the given values into the equation. The drug should be given about every 2.75 hours.