## Problem Set 26: Exponential and Logarithmic Equations

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?

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

4. ${4}^{-3v - 2}={4}^{-v}$

5. $64\cdot {4}^{3x}=16$

6. ${3}^{2x+1}\cdot {3}^{x}=243$

7. ${2}^{-3n}\cdot \frac{1}{4}={2}^{n+2}$

8. $625\cdot {5}^{3x+3}=125$

9. $\frac{{36}^{3b}}{{36}^{2b}}={216}^{2-b}$

10. ${\left(\frac{1}{64}\right)}^{3n}\cdot 8={2}^{6}$

For the following exercises, use logarithms to solve.

11. ${9}^{x - 10}=1$

12. $2{e}^{6x}=13$

13. ${e}^{r+10}-10=-42$

14. $2\cdot {10}^{9a}=29$

15. $-8\cdot {10}^{p+7}-7=-24$

16. $7{e}^{3n - 5}+5=-89$

17. ${e}^{-3k}+6=44$

18. $-5{e}^{9x - 8}-8=-62$

19. $-6{e}^{9x+8}+2=-74$

20. ${2}^{x+1}={5}^{2x - 1}$

21. ${e}^{2x}-{e}^{x}-132=0$

22. $7{e}^{8x+8}-5=-95$

23. $10{e}^{8x+3}+2=8$

24. $4{e}^{3x+3}-7=53$

25. $8{e}^{-5x - 2}-4=-90$

26. ${3}^{2x+1}={7}^{x - 2}$

27. ${e}^{2x}-{e}^{x}-6=0$

28. $3{e}^{3 - 3x}+6=-31$

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

29. $\mathrm{log}\left(\frac{1}{100}\right)=-2$

30. ${\mathrm{log}}_{324}\left(18\right)=\frac{1}{2}$

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

31. $5{\mathrm{log}}_{7}n=10$

32. $-8{\mathrm{log}}_{9}x=16$

33. $4+{\mathrm{log}}_{2}\left(9k\right)=2$

34. $2\mathrm{log}\left(8n+4\right)+6=10$

35. $10 - 4\mathrm{ln}\left(9 - 8x\right)=6$

For the following exercises, use the one-to-one property of logarithms to solve.

36. $\mathrm{ln}\left(10 - 3x\right)=\mathrm{ln}\left(-4x\right)$

37. ${\mathrm{log}}_{13}\left(5n - 2\right)={\mathrm{log}}_{13}\left(8 - 5n\right)$

38. $\mathrm{log}\left(x+3\right)-\mathrm{log}\left(x\right)=\mathrm{log}\left(74\right)$

39. $\mathrm{ln}\left(-3x\right)=\mathrm{ln}\left({x}^{2}-6x\right)$

40. ${\mathrm{log}}_{4}\left(6-m\right)={\mathrm{log}}_{4}3m$

41. $\mathrm{ln}\left(x - 2\right)-\mathrm{ln}\left(x\right)=\mathrm{ln}\left(54\right)$

42. ${\mathrm{log}}_{9}\left(2{n}^{2}-14n\right)={\mathrm{log}}_{9}\left(-45+{n}^{2}\right)$

43. $\mathrm{ln}\left({x}^{2}-10\right)+\mathrm{ln}\left(9\right)=\mathrm{ln}\left(10\right)$

For the following exercises, solve each equation for x.

44. $\mathrm{log}\left(x+12\right)=\mathrm{log}\left(x\right)+\mathrm{log}\left(12\right)$

45. $\mathrm{ln}\left(x\right)+\mathrm{ln}\left(x - 3\right)=\mathrm{ln}\left(7x\right)$

46. ${\mathrm{log}}_{2}\left(7x+6\right)=3$

47. $\mathrm{ln}\left(7\right)+\mathrm{ln}\left(2 - 4{x}^{2}\right)=\mathrm{ln}\left(14\right)$

48. ${\mathrm{log}}_{8}\left(x+6\right)-{\mathrm{log}}_{8}\left(x\right)={\mathrm{log}}_{8}\left(58\right)$

49. $\mathrm{ln}\left(3\right)-\mathrm{ln}\left(3 - 3x\right)=\mathrm{ln}\left(4\right)$

50. ${\mathrm{log}}_{3}\left(3x\right)-{\mathrm{log}}_{3}\left(6\right)={\mathrm{log}}_{3}\left(77\right)$

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. ${\mathrm{log}}_{9}\left(x\right)-5=-4$

52. ${\mathrm{log}}_{3}\left(x\right)+3=2$

53. $\mathrm{ln}\left(3x\right)=2$

54. $\mathrm{ln}\left(x - 5\right)=1$

55. $\mathrm{log}\left(4\right)+\mathrm{log}\left(-5x\right)=2$

56. $-7+{\mathrm{log}}_{3}\left(4-x\right)=-6$

57. $\mathrm{ln}\left(4x - 10\right)-6=-5$

58. $\mathrm{log}\left(4 - 2x\right)=\mathrm{log}\left(-4x\right)$

59. ${\mathrm{log}}_{11}\left(-2{x}^{2}-7x\right)={\mathrm{log}}_{11}\left(x - 2\right)$

60. $\mathrm{ln}\left(2x+9\right)=\mathrm{ln}\left(-5x\right)$

61. ${\mathrm{log}}_{9}\left(3-x\right)={\mathrm{log}}_{9}\left(4x - 8\right)$

62. $\mathrm{log}\left({x}^{2}+13\right)=\mathrm{log}\left(7x+3\right)$

63. $\frac{3}{{\mathrm{log}}_{2}\left(10\right)}-\mathrm{log}\left(x - 9\right)=\mathrm{log}\left(44\right)$

64. $\mathrm{ln}\left(x\right)-\mathrm{ln}\left(x+3\right)=\mathrm{ln}\left(6\right)$

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

65. An account with an initial deposit of \$6,500 earns 7.25% annual interest, compounded continuously. How much will the account be worth after 20 years?

66. The formula for measuring sound intensity in decibels D is defined by the equation $D=10\mathrm{log}\left(\frac{I}{{I}_{0}}\right)$, where I is the intensity of the sound in watts per square meter and ${I}_{0}={10}^{-12}$ is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of $8.3\cdot {10}^{2}$ watts per square meter?

67. The population of a small town is modeled by the equation $P=1650{e}^{0.5t}$ where t is measured in years. In approximately how many years will the town’s population reach 20,000?

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate x to 3 decimal places.

68. $1000{\left(1.03\right)}^{t}=5000$ using the common log.

69. ${e}^{5x}=17$ using the natural log

70. $3{\left(1.04\right)}^{3t}=8$ using the common log

71. ${3}^{4x - 5}=38$ using the common log

72. $50{e}^{-0.12t}=10$ using the natural log

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

73. $7{e}^{3x - 5}+7.9=47$

74. $\mathrm{ln}\left(3\right)+\mathrm{ln}\left(4.4x+6.8\right)=2$

75. $\mathrm{log}\left(-0.7x - 9\right)=1+5\mathrm{log}\left(5\right)$

76. Atmospheric pressure P in pounds per square inch is represented by the formula $P=14.7{e}^{-0.21x}$, where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5280 feet in a mile)

77. The magnitude M of an earthquake is represented by the equation $M=\frac{2}{3}\mathrm{log}\left(\frac{E}{{E}_{0}}\right)$ where E is the amount of energy released by the earthquake in joules and ${E}_{0}={10}^{4.4}$ is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing $1.4\cdot {10}^{13}$ joules of energy?

78. Use the definition of a logarithm along with the one-to-one property of logarithms to prove that ${b}^{{\mathrm{log}}_{b}x}=x$.

79. Recall the formula for continually compounding interest, $y=A{e}^{kt}$. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm.

80. Recall the compound interest formula $A=a{\left(1+\frac{r}{k}\right)}^{kt}$. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t.

81. Newton’s Law of Cooling states that the temperature T of an object at any time t can be described by the equation $T={T}_{s}+\left({T}_{0}-{T}_{s}\right){e}^{-kt}$, where ${T}_{s}$ is the temperature of the surrounding environment, ${T}_{0}$ is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm.