Exponential and Logarithmic Equations and Models: Get Stronger Answer Key

Logarithmic Properties

  1. [latex]\log_b(2)+\log_b(7)+\log_b(x)+\log_b(y)[/latex]
  2. [latex]\log_b(13)-\log_b(17)[/latex]
  3. [latex]-k\ln(4)[/latex]
  4. [latex]\ln(7xy)[/latex]
  5. [latex]\log_b(4)[/latex]
  6. [latex]\log_b(7)[/latex]
  7. [latex]15\log(x)+13\log(y)-19\log(z)[/latex]
  8. [latex]\dfrac{3}{2}\log(x)-2\log(y)[/latex]
  9. [latex]\dfrac{8}{3}\log(x)+\dfrac{14}{3}\log(y)[/latex]
  10. [latex]\ln(2x^7)[/latex]
  11. [latex]\log(\dfrac{xz^3}{\sqrt{y}})[/latex]
  12. [latex]\log_7(15)=\dfrac{\ln(15)}{\ln(7)}[/latex]
  13. [latex]\log_{11}(5)=\dfrac{\log_5(5)}{\log_5(11)}=\dfrac{1}{b}[/latex]
  14. [latex]\log_{11}(\dfrac{6}{11})=\dfrac{\log_5(\dfrac{6}{11})}{\log_5(11)}=\dfrac{\log_5(6)-\log_5(11)}{\log_5(11)}=\dfrac{a-b}{b}=\dfrac{a}{b}-1[/latex]
  15. [latex]3[/latex]
  16. [latex]2.81359[/latex]
  17. [latex]0.93913[/latex]
  18. [latex]-2.23266[/latex]

Exponential and Logarithmic Equations and Models

  1. [latex]x=-\dfrac{1}{3}[/latex]
  2. [latex]n=-1[/latex]
  3. [latex]b=\dfrac{6}{5}[/latex]
  4. [latex]x=10[/latex]
  5. No solution
  6. [latex]p=\log(\dfrac{17}{8})-7[/latex]
  7. [latex]k=-\dfrac{\ln(38)}{3}[/latex]
  8. [latex]x=\dfrac{\ln(\dfrac{38}{3})-8}{9}[/latex]
  9. [latex]x=\ln(12)[/latex]
  10. [latex]10^{-2}=\dfrac{1}{100}[/latex]
  11. [latex]n=49[/latex]
  12. [latex]k=\dfrac{1}{36}[/latex]
  13. [latex]x=\dfrac{9-e}{8}[/latex]
  14. [latex]n=1[/latex]
  15. No solution
  16. No solution
  17. [latex]x=\pm\dfrac{10}{3}[/latex]
  18. [latex]x=10[/latex]
  19. [latex]x=0[/latex]
  20. [latex]x=\dfrac{3}{4}[/latex]

Exponential and Logarithmic Models

  1. [latex]f(0)\approx 16.7[/latex]; The amount initially present is about [latex]16.7[/latex] units.
  2. [latex]150[/latex]
  3. exponential; [latex]f(x)=1.2^x[/latex]
  4. logarithmic
    Scatter plot of points representing a function f(x) with values increasing from x=1 to x=10. The points form an upward trend, with f(x) ranging from approximately 1 to 9. The graph includes labeled axes.
  5. logarithmic
    Scatter plot of points showing an upward trend with values on the y-axis increasing from approximately 9 to 13 as x progresses from 4 to 13. The points are closely spaced, indicating a steady increase, and the graph includes labeled axes.
  6. Graph of P(t)=1000/(1+9e^(-0.6t))
  7. about [latex]1.4[/latex] years
  8. about [latex]7.3[/latex] years
  9. [latex]A=125e^{(-0.3567t)}[/latex]; [latex]A\approx 43[/latex] mg
  10. about [latex]60[/latex] days
  11. [latex]A(t)=250e^{(-0.00822t)}[/latex]; half-life: about [latex]84[/latex] minutes
  12. [latex]r\approx -0.0667[/latex], So the hourly decay rate is about [latex]6.67 \%[/latex]
  13. [latex]f(t)=1350e^{(0.03466t)}[/latex]; after [latex]3[/latex] hours: [latex]P(180)\approx 691,200[/latex]
  14. [latex]f(t)=256e^{(0.068110t)}[/latex]; doubling time: about [latex]10[/latex] minutes
  15. about [latex]88[/latex] minutes
  16. [latex]T(t)=90e^{(-0.008377t)}+75[/latex], where [latex]t[/latex] is in minutes.
  17. about [latex]113[/latex] minutes