sectioN exercises 535

6.6 SeCTIOn exeRCISeS

veRbAl

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)

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight

dmabine

Highlight