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 5.5 Exponential and Logarithmic Equations and Inequalities Type I Exponential Equations –Solved in Section 5.2 –Easily written as powers of same base i.e. 125 x = 5 x Properties of Logarithmic and Exponential Functions For b > 0 and b  1: 1. if and only if x = y. 2.If x > 0 and y > 0, log b x = log b y if and only if x = y.

4. Copyright © 2007 Pearson Education, Inc. Slide 5-4 Type 2 Exponential Equations –Cannot be easily written as powers of same base i.e 7 x = 12 –General strategy: take the logarithm of both sides and apply the power rule to eliminate variable exponents ExampleSolve 7 x = 12. Solution 5.5 Type 2 Exponential Equations

5. Copyright © 2007 Pearson Education, Inc. Slide 5-5 5.5 Solving a Type 2 Exponential Inequality ExampleSolve 7 x < 12. SolutionFrom the previous example, 7 x = 12 when x  1.277. Using the graph below, y 1 = 7 x is below the graph y 2 = 12 for all x-values less than 1.277. The solution set is (– ,1.277).

6. Copyright © 2007 Pearson Education, Inc. Slide 5-6 5.5 Solving a Type 2 Exponential Equation Example Solve Solution Take logarithms of both sides. Apply the power rule. Distribute. Get all x-terms on one side. Factor out x and solve.

7. Copyright © 2007 Pearson Education, Inc. Slide 5-7 5.5 Solving a Logarithmic Equation of the Type log x = log y ExampleSolve Analytic SolutionThe domain must satisfy x + 6 > 0, x + 2 > 0, and x > 0. The intersection of these is (0,  ). Quotient property of logarithms log x = log y  x = y

8. Copyright © 2007 Pearson Education, Inc. Slide 5-8 5.5 Solving a Logarithmic Equation of the Type log x = log y Since the domain of the original equation was (0,  ), x = –3 cannot be a solution. The solution set is {2}. Multiply by x + 2. Solve the quadratic equation.

9. Copyright © 2007 Pearson Education, Inc. Slide 5-9 5.5 Solving a Logarithmic Equation of the Type log x = log y Graphing Calculator Solution The point of intersection is at x = 2. Notice that the graphs do not intersect at x = –3, thus supporting our conclusion that –3 is an extraneous solution.

10. Copyright © 2007 Pearson Education, Inc. Slide 5-10 5.5 Solving a Logarithmic Equation of the Type log x = k ExampleSolve Solution Since it is not in the domain and must be discarded, giving the solution set Write in exponential form.

11. Copyright © 2007 Pearson Education, Inc. Slide 5-11 5.5 Solving Equations Involving both Exponentials and Logarithms ExampleSolve SolutionThe domain is (0,  ). – 4 is not valid since – 4 0.

12. Copyright © 2007 Pearson Education, Inc. Slide 5-12 5.5 Solving Exponential and Logarithmic Equations An exponential or logarithmic equation can be solved by changing the equation into one of the following forms, where a and b are real numbers, a > 0, and a  1. 1.a f(x) = b Solve by taking the logarithm of each side. 2.log a f (x) = log a g (x) Solve f (x) = g (x) analytically. 3. log a f (x) = b Solve by changing to exponential form f (x) = a b.

13. Copyright © 2007 Pearson Education, Inc. Slide 5-13 5.5 Solving a Logarithmic Formula from Biology ExampleThe formula gives the number of species in a sample, where n is the number of individuals in the sample, and a is a constant indicating diversity. Solve for n. SolutionIsolate the logarithm and change to exponential form.