490 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions CAUTION Do not confuse a quotient like ln 12 ln 7 in Example 1 with ln 12 7 , which can be written as ln 12 - ln 7. We cannot change the quotient of two logarithms to a difference of logarithms. ln 12 ln 7 ≠ln 12 7 −20 −2 15 5 As seen in the display at the bottom of the screen, when rounded to three decimal places, the solution of 7x - 12 = 0 agrees with that found in Example 1. EXAMPLE 1 Solving an Exponential Equation Solve 7x = 12. Give the solution to the nearest thousandth. SOLUTION The properties of exponents cannot be used to solve this equation, so we apply the preceding property of logarithms. While any appropriate base b can be used, the best practical base is base 10 or base e. We choose base e (natural) logarithms here. 7x = 12 l n 7x = ln 12 Property of logarithms x ln 7 = ln 12 Power property x = ln 12 ln 7 Divide by ln 7. x ≈1.277 Use a calculator. This is approximate. This is exact. The solution set is 51.2776. S Now Try Exercise 11. EXAMPLE 2 Solving an Exponential Equation Solve 32x-1 = 0.4x+2. Give the solution to the nearest thousandth. SOLUTION 32x-1 = 0.4x+2 ln 32x-1 = ln 0.4x+2 Take the natural logarithm on each side. 12x - 12 ln 3 = 1x + 22 ln 0.4 Power property 2x ln 3 - ln 3 = x ln 0.4 + 2 ln 0.4 Distributive property 2x ln 3 - x ln 0.4 = 2 ln 0.4 + ln 3 Write so that the terms with x are on one side. x12 ln 3 - ln 0.42 = 2 ln 0.4 + ln 3 Factor out x. x = 2 ln 0.4 + ln 3 2 ln 3 - ln 0.4 Divide by 2 ln 3 - ln 0.4. x = ln 0.42 + ln 3 ln 32 - ln 0.4 Power property x = ln 0.16 + ln 3 ln 9 - ln 0.4 Apply the exponents. x = ln 0.48 ln 22.5 Product and quotient properties x ≈ -0.236 Use a calculator. This is exact. This is approximate. The solution set is 5-0.2366. S Now Try Exercise 19. −3 −4 3 4 This screen supports the solution found in Example 2.
RkJQdWJsaXNoZXIy NjM5ODQ=