Solutions 5.6: Exponential and Logarithmic Equations

Solutions to Try Its

1. x = –2

2. x = –1

3. [latex]x=\frac{1}{2}[/latex]

4. The equation has no solution.

5. [latex]x=\frac{\mathrm{ln}3}{\mathrm{ln}}\left(23\right)[/latex]

6. [latex]t=2\mathrm{ln}\left(\frac{11}{3}\right)[/latex] or [latex]\mathrm{ln}{\left(\frac{11}{3}\right)}^{2}[/latex]

7. [latex]t=\mathrm{ln}\left(\frac{1}{\sqrt{2}}\right)=-\frac{1}{2}\mathrm{ln}\left(2\right)[/latex]

8. [latex]x=\mathrm{ln}2[/latex]

9. [latex]x={e}^{4}[/latex]

10. [latex]x={e}^{5}-1[/latex]

11. [latex]x\approx 9.97[/latex]

12. x = 1 or x = –1

13. [latex]t=703,800,000\times \frac{\mathrm{ln}\left(0.8\right)}{\mathrm{ln}\left(0.5\right)}\text{ years }\approx \text{ }226,572,993\text{ years}[/latex].

Solutions to Odd-Numbered Exercises

1. Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.

3. The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.

5. [latex]x=-\frac{1}{3}[/latex]

7. n = –1

9. [latex]b=\frac{6}{5}[/latex]

11. x = 10

13. No solution

15. [latex]p=\mathrm{log}\left(\frac{17}{8}\right)-7[/latex]

17. [latex]k=-\frac{\mathrm{ln}\left(38\right)}{3}[/latex]

19. [latex]x=\frac{\mathrm{ln}\left(\frac{38}{3}\right)-8}{9}[/latex]

21. [latex]x=\mathrm{ln}12[/latex]

23. [latex]x=\frac{\mathrm{ln}\left(\frac{3}{5}\right)-3}{8}[/latex]

25. no solution

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

29. [latex]{10}^{-2}=\frac{1}{100}[/latex]

31. n = 49

33. [latex]k=\frac{1}{36}[/latex]

35. [latex]x=\frac{9-e}{8}[/latex]

37. n = 1

39. No solution

41. No solution

43. [latex]x=\pm \frac{10}{3}[/latex]

45. x = 10

47. x = 0

49. [latex]x=\frac{3}{4}[/latex]

51. x = 9
Graph of log_9(x)-5=y and y=-4.

53. [latex]x=\frac{{e}^{2}}{3}\approx 2.5[/latex]
Graph of ln(3x)=y and y=2.

55. x = –5
Graph of log(4)+log(-5x)=y and y=2.

57. [latex]x=\frac{e+10}{4}\approx 3.2[/latex]
Graph of ln(4x-10)-6=y and y=-5.

59. No solution
Graph of log_11(-2x^2-7x)=y and y=log_11(x-2).

61. [latex]x=\frac{11}{5}\approx 2.2[/latex]
Graph of log_9(3-x)=y and y=log_9(4x-8).

63. [latex]x=\frac{101}{11}\approx 9.2[/latex]

65. about $27,710.24

67. about 5 years

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

71. [latex]x=\frac{\mathrm{log}\left(38\right)+5\mathrm{log}\left(3\right)\text{ }}{4\mathrm{log}\left(3\right)}\approx 2.078[/latex]

73. [latex]x\approx 2.2401[/latex]

75. [latex]x\approx -44655.7143[/latex]

77. about 5.83

79. [latex]t=\mathrm{ln}\left({\left(\frac{y}{A}\right)}^{\frac{1}{k}}\right)[/latex]

81. [latex]t=\mathrm{ln}\left({\left(\frac{T-{T}_{s}}{{T}_{0}-{T}_{s}}\right)}^{-\frac{1}{k}}\right)[/latex]