#### Transcript Ch 5 Inverse, Exponential and Logarithmic Functions

Chapter 5: Exponential and Logarithmic Functions 5.1 5.2 5.3 5.4 5.5 Inverse Functions Exponential Functions Logarithms and Their Properties Logarithmic Functions Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions Copyright © 2007 Pearson Education, Inc. Slide 5-2 5.5 Exponential and Logarithmic Equations and Inequalities Properties of Logarithmic and Exponential Functions For b > 0 and b 1: 1. b x b y if and only if x = y. 2. If x > 0 and y > 0, logb x = logb y if and only if x = y. • Type I Exponential Equations – Solved in Section 5.2 – Easily written as powers of same base i.e. 125x = 5x Copyright © 2007 Pearson Education, Inc. Slide 5-3 5.5 Type 2 Exponential Equations • Type 2 Exponential Equations – Cannot be easily written as powers of same base i.e 7x = 12 – General strategy: take the logarithm of both sides and apply the power rule to eliminate variable exponents Example Solve 7x = 12. Solution 7 x 12 ln 7 x ln 12 ln 12 x ln 7 ln 12 x 1.277 ln 7 Copyright © 2007 Pearson Education, Inc. Slide 5-4 5.5 Solving a Type 2 Exponential Inequality Example Solve 7x < 12. Solution From the previous example, 7x = 12 when x 1.277. Using the graph below, y1 = 7x is below the graph y2 = 12 for all x-values less than 1.277. The solution set is (–,1.277). Copyright © 2007 Pearson Education, Inc. Slide 5-5 5.5 Solving a Type 2 Exponential Equation Example 3 x 1 4 x 2 3 . Solve Solution 23 x 1 34 x log 23 x 1 log 34 x (3 x 1) log 2 (4 x) log 3 3 x log 2 log 2 4 log 3 x log 3 3 x log 2 x log 3 4 log 3 log 2 x(3 log 2 log 3) 4 log 3 log 2 4 log 3 log 2 x 3 log 2 log 3 Copyright © 2007 Pearson Education, Inc. Take logarithms of both sides. Apply the power rule. Distribute. Get all x-terms on one side. Factor out x and solve. Slide 5-6 5.5 Solving a Logarithmic Equation of the Type log x = log y Example Solve log3 ( x 6) log3 ( x 2) log3 x. Analytic Solution The domain must satisfy x + 6 > 0, x + 2 > 0, and x > 0. The intersection of these is (0,). log 3 ( x 6) log 3 ( x 2) log 3 x x6 log 3 log 3 x x2 x6 x x2 Copyright © 2007 Pearson Education, Inc. Quotient property of logarithms log x = log y x = y Slide 5-7 5.5 Solving a Logarithmic Equation of the Type log x = log y x 6 x ( x 2) Multiply by x + 2. x 6 x2 2x 0 x x6 0 ( x 3)( x 2) 2 x 3 or Solve the quadratic equation. x2 Since the domain of the original equation was (0,), x = –3 cannot be a solution. The solution set is {2}. Copyright © 2007 Pearson Education, Inc. Slide 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. Copyright © 2007 Pearson Education, Inc. Slide 5-9 5.5 Solving a Logarithmic Equation of the Type log x = k Example Solve log( 3x 2) log( x 1) 1. Solution log(3 x 2) log( x 1) 1 log(3 x 2)( x 1) 1 Write in exponential form. (3 x 2)( x 1) 101 3 x 2 x 2 10 3 x 2 x 12 0 1 145 x 6 Since 1 6145 1, it is not in the domain and must be discarded, giving the solution set 1 6145 1. Copyright © 2007 Pearson Education, Inc. Slide 5-10 5.5 Solving Equations Involving both Exponentials and Logarithms Example Solve e2 ln x 161 . Solution The domain is (0,). e 2 ln x e ln x 2 x 2 x 2 1 16 1 16 1 16 42 Power rule e ln u u – 4 is not valid since – 4 < 0, and x > 0. x4 Copyright © 2007 Pearson Education, Inc. Slide 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. loga f (x) = loga g (x) Solve f (x) = g (x) analytically. 3. loga f (x) = b Solve by changing to exponential form f (x) = ab. Copyright © 2007 Pearson Education, Inc. Slide 5-12 5.5 Solving a Logarithmic Formula from Biology Example The formula S a ln 1 an 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. Solution Isolate the logarithm and change to exponential form. S n ln 1 a a n e 1 a n a (e 1) S a S Copyright © 2007 Pearson Education, Inc. a Slide 5-13