We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > Precalculus I

Solutions 26: 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]

Licenses & Attributions

CC licensed content, Shared previously

  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..