1. Copyright © 2011 Pearson Education, Inc. Slide 5.5-1 5.5 Exponential and Logarithmic Equations and Inequalities 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, then log b x = log b y if and only if x = y.

2. Copyright © 2011 Pearson Education, Inc. Slide 5.5-2 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 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.

3. Copyright © 2011 Pearson Education, Inc. Slide 5.5-3 ExampleSolve 7 x = 12. Solution 5.5 Type 2 Exponential Equations

4. Copyright © 2011 Pearson Education, Inc. Slide 5.5-4 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).

5. Copyright © 2011 Pearson Education, Inc. Slide 5.5-5 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.

6. Copyright © 2011 Pearson Education, Inc. Slide 5.5-6 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

7. Copyright © 2011 Pearson Education, Inc. Slide 5.5-7 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.

8. Copyright © 2011 Pearson Education, Inc. Slide 5.5-8 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.

9. Copyright © 2011 Pearson Education, Inc. Slide 5.5-9 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.

10. Copyright © 2011 Pearson Education, Inc. Slide 5.5-10 5.5 Solving Equations Involving both Exponentials and Logarithms ExampleSolve SolutionThe domain is (0,  ). – 4 is not valid since – 4 0.

11. Copyright © 2011 Pearson Education, Inc. Slide 5.5-11 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.

12. Copyright © 2011 Pearson Education, Inc. Slide 5.5-12 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.