466 6.4.1 Exponential and Logarithmic Functions Exercises In Exercises 1 - 24, solve the equation analytically. 1. log(3x 1) = log(4 x) 2. log2 x3 = log2 (x) 3. ln 8 x2 = ln(2 x) 4. log5 18 x2 = log5 (6 x) 5. log3 (7 2x) = 2 6. log 1 (2x 1) = 3 7. ln x2 99 = 0 8. log(x2 3x) = 1 9. log125 3x 2 2x + 3 2 = 1 3 10. log x 103 = 4.7 x 1012 = 150 11. log(x) = 5.4 12. 10 log 13. 6 3 log5 (2x) = 0 14. 3 ln(x) 2 = 1 ln(x) 15. log3 (x 4) + log3 (x + 4) = 2 16. log5 (2x + 1) + log5 (x + 2) = 1 17. log169 (3x + 7) log169 (5x 9) = 1 2 18. ln(x + 1) ln(x) = 3 19. 2 log7 (x) = log7 (2) + log7 (x + 12) 20. log(x) log(2) = log(x + 8) log(x + 2) 21. log3 (x) = log 1 (x) + 8 22. ln(ln(x)) = 3 23. (log(x))2 = 2 log(x) + 15 24. ln(x2 ) = (ln(x))2 3 In Exercises 25 - 30, solve the inequality analytically. 1 ln(x) <0 x2 x 27. 10 log 90 1012 25. 29. 2.3 < log(x) < 5.4 26. x ln(x) x > 0 28. 5.6 log x 103 7.1 30. ln(x2 ) (ln(x))2 In Exercises 31 - 34, use your calculator to help you solve the equation or inequality. 4 31. ln(x) = ex 32. ln(x) = 33. ln(x2 + 1) 5 34. ln(2x3 x2 + 13x 6) < 0 x 6.4 Logarithmic Equations and Inequalities 467 35. Since f (x) = ex is a strictly increasing function, if a < b then ea < eb . Use this fact to solve the inequality ln(2x + 1) < 3 without a sign diagram. Use this technique to solve the inequalities in Exercises 27 - 29. (Compare this to Exercise 46 in Section 6.3.) 36. Solve ln(3 y) ln(y) = 2x + ln(5) for y. 37. In Example 6.4.4 we found the inverse of f (x) = x log(x) to be f 1 (x) = 10 x+1 . 1 log(x) (a) Show that f 1 f (x) = x for all x in the domain of f and that f f 1 (x) = x for all x in the domain of f 1 . (b) Find the range of f by nding the domain of f 1 . x (c) Let g(x) = and h(x) = log(x). Show that f = g h and (g h)1 = h1 g 1 . 1x (We know this is true in general by Exercise 31 in Section 5.2, but it's nice to see a specic example of the property.) 38. Let f (x) = 1 ln 2 1+x . Compute f 1 (x) and nd its domain and range. 1x 39. Explain the equation in Exercise 10 and the inequality in Exercise 28 above in terms of the Richter scale for earthquake magnitude. (See Exercise 75 in Section 6.1.) 40. Explain the equation in Exercise 12 and the inequality in Exercise 27 above in terms of sound intensity level as measured in decibels. (See Exercise 76 in Section 6.1.) 41. Explain the equation in Exercise 11 and the inequality in Exercise 29 above in terms of the pH of a solution. (See Exercise 77 in Section 6.1.) 42. With the help of your classmates, solve the inequality n x > ln(x) for a variety of natural numbers n. What might you conjecture about the \"speed\" at which f (x) = ln(x) grows versus any principal nth root function? 468 Exponential and Logarithmic Functions 6.4.2 Answers 1. x = 5 4 2. x = 1 3. x = 2 4. x = 3, 4 5. x = 1 6. x = 7. x = 10 8. x = 2, 5 9. x = 17 7 10. x = 101.7 11. x = 105.4 12. x = 103 9 2 13. x = 25 2 14. x = e3/4 15. x = 5 16. x = 1 2 17. x = 2 18. x = 19. x = 6 20. x = 4 21. x = 81 22. x = ee 23. x = 103 , 105 24. x = 1, x = e2 25. (e, ) 26. (e, ) 27. 103 , 28. 102.6 , 104.1 29. 105.4 , 102.3 30. (0, 1] [e2 , ) 3 1 e3 1 31. x 1.3098 32. x 4.177, x 5503.665 33. (, 12.1414) (12.1414, ) 34. (3.0281, 3)(0.5, 0.5991)(1.9299, 2) 1 e3 1 35. < x < 2 2 36. y = e2x ex ex 3 5e2x +1 1 = x . (To see why we rewrite this in this form, see Exercise 51 in +1 e + ex Section 11.10.) The domain of f 1 is (, ) and its range is the same as the domain of f , namely (1, 1). 38. f 1 (x) = e2x