496 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions (b) Replace ƒ1t2 with 130 and solve for t. ƒ1t2 = 23.19 ln t + 75.19 Given function 130 = 23.19 ln t + 75.19 Let ƒ1t2 = 130. 54.81 = 23.19 ln t Subtract 75.19. ln t = 54.81 23.19 Divide by 23.19 and rewrite. t = e54.81/23.19 Write in exponential form. t ≈10.63 Use a calculator. Adding 10.63 to 2012 gives 2022.63. Based on this model, the number of U.S. residents who watch video content on a tablet will reach 130 million in 2022. S Now Try Exercise 107. 4.5 Exercises CONCEPT PREVIEW Match each equation in Column I with the best first step for solving it in Column II. I 1. 10x = 150 2. e2x-1 = 24 3. log4 1x 2 - 102 = 2 4. e2x # ex = 2e 5. 2e2x - 5ex - 3 = 0 6. log 12x - 12 + log 1x + 42 = 1 II A. Use the product rule for exponents. B. Take the common logarithm on each side. C. Write the sum of logarithms as the logarithm of a product. D. Let u = ex and write the equation in quadratic form. E. Change to exponential form. F. Take the natural logarithm on each side. CONCEPT PREVIEW An exponential equation such as 5x = 9 can be solved for its exact solution using the meaning of logarithm and the change-ofbase theorem. Because x is the exponent to which 5 must be raised in order to obtain 9, the exact solution is log5 9, or log 9 log 5 , or ln 9 ln 5 . For each equation, give the exact solution in three forms similar to the forms above. 7. 7x = 19 8. 3x = 10 9. a 1 2b x = 12 10. a 1 3b x = 4 Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35–40, give solutions in exact form. See Examples 1–4. 11. 3x = 7 12. 5x = 13 13. a 1 2b x = 5 14. a 1 3b x = 6 15. 0.8x = 4 16. 0.6x = 3 17. 4x-1 = 32x 18. 2x+3 = 52x 19. 6x+1 = 42x-1
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