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?

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

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. $3{e}^{3-3x}+6=-31$

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

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

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

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

44. $x=±\frac{10}{3}$

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

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

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

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

49. $x=0$

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

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

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

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

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

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

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

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

58. $-7+{\mathrm{log}}_{3}\left(4-x\right)=-6$

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

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

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

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

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

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

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

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

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

68. The formula for measuring sound intensity in decibels$\text{ }D\text{ }$is defined by the equation$\text{ }D=10\mathrm{log}\left(\frac{I}{{I}_{0}}\right),\text{}$where$\text{ }I\text{ }$is the intensity of the sound in watts per square meter and$\text{ }{I}_{0}={10}^{-12}\text{ }$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$\text{ }8.3\cdot {10}^{2}\text{ }$watts per square meter?

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

70. $1000{\left(1.03\right)}^{t}=5000\text{ }$using the common log.

71. ${e}^{5x}=17\text{ }$using the natural log

72. $\frac{\mathrm{ln}\left(17\right)}{5}\approx 0.567$

73. $3{\left(1.04\right)}^{3t}=8\text{ }$using the common log

74. ${3}^{4x-5}=38\text{ }$using the common log

75. $50{e}^{-0.12t}=10\text{ }$using the natural log

76. $7{e}^{3x-5}+7.9=47$

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

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

79. Atmospheric pressure$\text{ }P\text{ }$in pounds per square inch is represented by the formula$\text{ }P=14.7{e}^{-0.21x},$ 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$\text{ }8.369\text{ }$pounds per square inch? (Hint: there are 5280 feet in a mile)

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

about$\text{ }5.83$

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

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

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

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