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. [latex]{4}^{-3v-2}={4}^{-v}[/latex]

5. [latex]64\cdot {4}^{3x}=16[/latex]

6. [latex]{3}^{2x+1}\cdot {3}^{x}=243[/latex]

7. [latex]{2}^{-3n}\cdot \frac{1}{4}={2}^{n+2}[/latex]

8. [latex]625\cdot {5}^{3x+3}=125[/latex]

9. [latex]\frac{{36}^{3b}}{{36}^{2b}}={216}^{2-b}[/latex]

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

11. [latex]{9}^{x-10}=1[/latex]

12. [latex]2{e}^{6x}=13[/latex]

13. [latex]{e}^{r+10}-10=-42[/latex]

14. [latex]2\cdot {10}^{9a}=29[/latex]

15. [latex]-8\cdot {10}^{p+7}-7=-24[/latex]

16. [latex]7{e}^{3n-5}+5=-89[/latex]

17. [latex]{e}^{-3k}+6=44[/latex]

18. [latex]-5{e}^{9x-8}-8=-62[/latex]

19. [latex]-6{e}^{9x+8}+2=-74[/latex]

20. [latex]{2}^{x+1}={5}^{2x-1}[/latex]

21. [latex]{e}^{2x}-{e}^{x}-132=0[/latex]

22. [latex]7{e}^{8x+8}-5=-95[/latex]

23. [latex]10{e}^{8x+3}+2=8[/latex]

24. [latex]4{e}^{3x+3}-7=53[/latex]

25. [latex]8{e}^{-5x-2}-4=-90[/latex]

26. [latex]{3}^{2x+1}={7}^{x-2}[/latex]

27. [latex]{e}^{2x}-{e}^{x}-6=0[/latex]

28. [latex]3{e}^{3-3x}+6=-31[/latex]

29. [latex]\mathrm{log}\left(\frac{1}{100}\right)=-2[/latex]

30. [latex]{\mathrm{log}}_{324}\left(18\right)=\frac{1}{2}[/latex]

31. [latex]5{\mathrm{log}}_{7}n=10[/latex]

32. [latex]-8{\mathrm{log}}_{9}x=16[/latex]

33. [latex]4+{\mathrm{log}}_{2}\left(9k\right)=2[/latex]

34. [latex]2\mathrm{log}\left(8n+4\right)+6=10[/latex]

35. [latex]10-4\mathrm{ln}\left(9-8x\right)=6[/latex]

36. [latex]\mathrm{ln}\left(10-3x\right)=\mathrm{ln}\left(-4x\right)[/latex]

37. [latex]{\mathrm{log}}_{13}\left(5n-2\right)={\mathrm{log}}_{13}\left(8-5n\right)[/latex]

38. [latex]\mathrm{log}\left(x+3\right)-\mathrm{log}\left(x\right)=\mathrm{log}\left(74\right)[/latex]

39. [latex]\mathrm{ln}\left(-3x\right)=\mathrm{ln}\left({x}^{2}-6x\right)[/latex]

40. [latex]{\mathrm{log}}_{4}\left(6-m\right)={\mathrm{log}}_{4}3m[/latex]

41. [latex]\mathrm{ln}\left(x-2\right)-\mathrm{ln}\left(x\right)=\mathrm{ln}\left(54\right)[/latex]

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

43. [latex]\mathrm{ln}\left({x}^{2}-10\right)+\mathrm{ln}\left(9\right)=\mathrm{ln}\left(10\right)[/latex]

44. [latex]x=±\frac{10}{3}[/latex]

45. [latex]\mathrm{log}\left(x+12\right)=\mathrm{log}\left(x\right)+\mathrm{log}\left(12\right)[/latex]

46. [latex]\mathrm{ln}\left(x\right)+\mathrm{ln}\left(x-3\right)=\mathrm{ln}\left(7x\right)[/latex]

47. [latex]{\mathrm{log}}_{2}\left(7x+6\right)=3[/latex]

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

49. [latex]x=0[/latex]

50. [latex]{\mathrm{log}}_{8}\left(x+6\right)-{\mathrm{log}}_{8}\left(x\right)={\mathrm{log}}_{8}\left(58\right)[/latex]

51. [latex]\mathrm{ln}\left(3\right)-\mathrm{ln}\left(3-3x\right)=\mathrm{ln}\left(4\right)[/latex]

52. [latex]{\mathrm{log}}_{3}\left(3x\right)-{\mathrm{log}}_{3}\left(6\right)={\mathrm{log}}_{3}\left(77\right)[/latex]

53. [latex]{\mathrm{log}}_{9}\left(x\right)-5=-4[/latex]

54. [latex]{\mathrm{log}}_{3}\left(x\right)+3=2[/latex]

55. [latex]\mathrm{ln}\left(3x\right)=2[/latex]

56. [latex]\mathrm{ln}\left(x-5\right)=1[/latex]

57. [latex]\mathrm{log}\left(4\right)+\mathrm{log}\left(-5x\right)=2[/latex]

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

59. [latex]\mathrm{ln}\left(4x-10\right)-6=-5[/latex]

60. [latex]\mathrm{log}\left(4-2x\right)=\mathrm{log}\left(-4x\right)[/latex]

61. [latex]{\mathrm{log}}_{11}\left(-2{x}^{2}-7x\right)={\mathrm{log}}_{11}\left(x-2\right)[/latex]

62. [latex]\mathrm{ln}\left(2x+9\right)=\mathrm{ln}\left(-5x\right)[/latex]

63. [latex]{\mathrm{log}}_{9}\left(3-x\right)={\mathrm{log}}_{9}\left(4x-8\right)[/latex]

64. [latex]\mathrm{log}\left({x}^{2}+13\right)=\mathrm{log}\left(7x+3\right)[/latex]

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

66. [latex]\mathrm{ln}\left(x\right)-\mathrm{ln}\left(x+3\right)=\mathrm{ln}\left(6\right)[/latex]

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

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

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

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

71. [latex]{e}^{5x}=17\text{ }[/latex]using the natural log

72. [latex]\frac{\mathrm{ln}\left(17\right)}{5}\approx 0.567[/latex]

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

74. [latex]{3}^{4x-5}=38\text{ }[/latex]using the common log

75. [latex]50{e}^{-0.12t}=10\text{ }[/latex]using the natural log

76. [latex]7{e}^{3x-5}+7.9=47[/latex]

77. [latex]\mathrm{ln}\left(3\right)+\mathrm{ln}\left(4.4x+6.8\right)=2[/latex]

78. [latex]\mathrm{log}\left(-0.7x-9\right)=1+5\mathrm{log}\left(5\right)[/latex]

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

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

about[latex]\text{ }5.83[/latex]

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

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

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

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