sectioN exercises 535

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1. How can an exponential equation be solved? 2. When does an extraneous solution occur? How can an extraneous solution be recognized?

3. When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

AlgebRAIC For the following exercises, use like bases to solve the exponential equation.

4. 4−3v − 2 = 4−v 5. 64 ⋅ 43x = 16 6. 32x + 1 ⋅ 3x = 243

7. 2−3n ⋅ 1 _ 4 = 2 n + 2 8. 625 ⋅ 53x + 3 = 125 9.

363b _ 362b

= 216 2 − b

10. 1 _ 64 3n

⋅ 8 = 26

For the following exercises, use logarithms to solve. 11. 9x − 10 = 1 12. 2e 6x = 13 13. e r + 10 − 10 = −42

14. 2 ⋅ 109a = 29 15. −8 ⋅ 10 p + 7 − 7 = −24 16. 7e 3n − 5 + 5 = −89

17. e −3k + 6 = 44 18. −5e 9x − 8 − 8 = −62 19. −6e 9x + 8 + 2 = −74 20. 2x + 1 = 52x − 1 21. e 2x − e x − 132 = 0 22. 7e8x + 8 − 5 = −95

23. 10e 8x + 3 + 2 = 8 24. 4e 3x + 3 − 7 = 53 25. 8e−5x − 2 − 4 = −90

26. 32x + 1 = 7x − 2 27. e 2x − e x − 6 = 0 28. 3e 3 − 3x + 6 = −31

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

29. log 1 _ 100 = −2 30. log324(18) = 1 _ 2

For the following exercises, use the definition of a logarithm to solve the equation.

31. 5log7(n) = 10 32. −8log9(x) = 16 33. 4 + log2(9k) = 2 34. 2log(8n + 4) + 6 = 10 35. 10 − 4ln(9 − 8x) = 6

For the following exercises, use the one-to-one property of logarithms to solve. 36. ln(10 − 3x) = ln(−4x) 37. log13(5n − 2) = log13(8 − 5n) 38. log(x + 3) − log(x) = log(74)

39. ln(−3x) = ln(x2 − 6x) 40. log4(6 − m) = log43(m) 41. ln(x − 2) − ln(x) = ln(54)

42. log9(2n 2 − 14n)= log9(−45 + n

2) 43. ln(x2 − 10) + ln(9) = ln(10)

For the following exercises, solve each equation for x. 44. log(x + 12) = log(x) + log(12) 45. ln(x) + ln(x − 3) = ln(7x) 46. log2(7x + 6) = 3 47. ln(7) + ln(2 − 4×2) = ln(14) 48. log8(x + 6) − log8(x) = log8(58) 49. ln(3) − ln(3 − 3x) = ln(4) 50. log3(3x) − log3(6) = log3(77)

gRAPhICAl For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

51. log9(x) − 5 = −4 52. log3(x) + 3 = 2 53. ln(3x) = 2

54. ln(x − 5) = 1 55. log(4) + log(−5x) = 2 56. −7 + log3 (4 − x) = −6 57. ln(4x − 10) − 6 = −5 58. log(4 − 2x) = log(−4x) 59. log11(−2x

2 − 7x) = log11(x − 2)

60. ln(2x + 9) = ln(−5x) 61. log9(3 − x) = log9(4x − 8) 62. log(x 2 + 13) = log(7x + 3)

63. 3 _ log2(10)

− log(x − 9) = log(44) 64. ln(x) − ln(x + 3) = ln(6)

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