4.3E: Logarithmic Functions (Exercises)
- Page ID
- 13907
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)section 4.3 exercise
Rewrite each equation in exponential form
1. \(\log _{4} (q)=m\)
2. \(\log _{3} (t)=k\)
3. \(\log _{a} (b)=c\)
4. \(\log _{p} (z)=u\)
5.\(\log \left(v\right)=t\)
6. \(\log \left(r\right)=s\)
7. \(\ln \left(w\right)=n\)
8. \(\ln \left(x\right)=y\)
Rewrite each equation in logarithmic form.
9. \(4^{x} =y\)
10. \(5^{y} =x\)
11. \(c^{d} =k\)
12. \(n^{z} =L\)
13. \(10^{a} =b\)
14. \(10^{p} =v\)
15. \(e^{k} =h\)
16. \(e^{y} =x\)
Solve for \(x\).
17. \(\log _{3} \left(x\right)=2\)
18. \(\log _{4} (x)=3\)
19. \(\log _{2} (x)=-3\)
20. \(\log _{5} (x)=-1\)
21. \(\log \left(x\right)=3\)
22. \(\log \left(x\right)=5\)
23. \(\ln \left(x\right)=2\)
24. \(\ln \left(x\right)=-2\)
Simplify each expression using logarithm properties.
25. \(\log _{5} \left(25\right)\)
26. \(\log _{2} \left(8\right)\)
27. \(\log _{3} \left(\dfrac{1}{27} \right)\)
28. \(\log _{6} \left(\dfrac{1}{36} \right)\)
29. \(\log _{6} \left(\sqrt{6} \right)\)
30. \(\log _{5} \left(\sqrt[{3}]{5} \right)\)
31. \(\log \left(10,000\right)\)
32. \(\log \left(100\right)\)
33. \(\log \left(0.001\right)\)
34. \(\log \left(0.00001\right)\)
35. \(\ln \left(e^{-2} \right)\)
36. \(\ln \left(e^{3} \right)\)
Evaluate using your calculator.
37. \(\log \left(0.04\right)\)
38. \(\log \left(1045\right)\)
39. \(\ln \left(15\right)\)
40. \(\ln \left(0.02\right)\)
Solve each equation for the variable.
41. \(5^{x} =14\)
42. \(3^{x} =23\)
43. \(7^{x} =\dfrac{1}{15}\)
44. \(3^{x} =\dfrac{1}{4}\)
45. \(e^{5x} =17\)
46. \(e^{3x} =12\)
47. \(3^{4x-5} =38\)
48. \(4^{2x-3} =44\)
49. \(1000\left(1.03\right)^{t} =5000\)
50. \(200\left(1.06\right)^{t} =550\)
51. \(3\left(1.04\right)^{3t} =8\)
52. \(2\left(1.08\right)^{4t} =7\)
53. \(50e^{-0.12t} =10\)
54. \(10e^{-0.03t} =4\)
55. \(10-8\left(\dfrac{1}{2} \right)^{x} =5\)
56. \(100-100\left(\dfrac{1}{4} \right)^{x} =70\)
Convert the equation into continuous growth form, \(f\left(t\right)=ae^{kt}\).
57. \(f\left(t\right)=300\left(0.91\right)^{t}\)
58. \(f\left(t\right)=120\left(0.07\right)^{t}\)
59. \(f\left(t\right)=10\left(1.04\right)^{t}\)
60. \(f\left(t\right)=1400\left(1.12\right)^{t}\)
Convert the equation into annual growth form, \(f\left(t\right)=ab^{t}\).
61. \(f\left(t\right)=\; 150e^{0.06t}\)
62. \(f\left(t\right)=100e^{0.12t}\)
63. \(f\left(t\right)=50e^{-0.012t}\)
64. \(f\left(t\right)=80e^{-0.85t}\)
65. The population of Kenya was 39.8 million in 2009 and has been growing by about 2.6% each year. If this trend continues, when will the population exceed 45 million?
66. The population of Algeria was 34.9 million in 2009 and has been growing by about 1.5% each year. If this trend continues, when will the population exceed 45 million?
67. The population of Seattle grew from 563,374 in 2000 to 608,660 in 2010. If the population continues to grow exponentially at the same rate, when will the population exceed 1 million people?
68. The median household income (adjusted for inflation) in Seattle grew from $42,948 in 1990 to $45,736 in 2000. If it continues to grow exponentially at the same rate, when will median income exceed $50,000?
69. A scientist begins with 100 mg of a radioactive substance. After 4 hours, it has decayed to 80 mg. How long after the process began will it take to decay to 15 mg?
70. A scientist begins with 100 mg of a radioactive substance. After 6 days, it has decayed to 60 mg. How long after the process began will it take to decay to 10 mg?
71. If $1000 is invested in an account earning 3% compounded monthly, how long will it take the account to grow in value to $1500?
72. If $1000 is invested in an account earning 2% compounded quarterly, how long will it take the account to grow in value to $1300?
- Answer
-
1. \(4^m = q\)
3. \(a^c = b\)
5. \(10^t = v\)
7. \(e^n = w\)
9. \(\text{log}_{4} (y) = x\)
11. \(\text{log}_{c} (k) = d\)
13. \(\text{log} (b) = a\)
15. \(\text{ln} (h) = k\)
17. 9
19. 1/8
21. 1000
23. \(e^2\)
25. 2
27. -3
29. 1/2
31. 4
33. -3
35. -2
37. -1.398
39. 2.708
41. \(\dfrac{\text{log}(14)}{\text{log}(5)} \approx 1.6397\)
43. \(\dfrac{\text{log}(\dfrac{1}{15})}{\text{log}(7)} \approx -1.392\)
45. \(\dfrac{\text{ln}(17)}{5} \approx 1.6397\)
47. \(\dfrac{\dfrac{\text{log}(38)}{\text{log}(3)} + 5}{4} \approx 2.078\)
49. \(\dfrac{\text{log}(5)}{\text{log}(1.03)} \approx 54.449\)
51. \(\dfrac{\text{log}(\dfrac{8}{3})}{\text{3log}(1.04)} \approx 8.335\)
53. \(\dfrac{\text{ln}(\dfrac{1}{5})}{-0.12} \approx 1.6397\)
55. \(\dfrac{\text{log}(\dfrac{5}{8})}{\text{log}(\dfrac{1}{2})} \approx 1.6397\)
57. \(f(t) = 300e^{-0.0943t}\)
59. \(f(t) = 10e^{0.03922t}\)
61. \(f(t) = 150(1.0618)^t\)
63. \(f(t) = 50 (0.98807)^t\)
65. During the year 2013
67. During the year 2074
69. \(\approx\) 34 \(hours\)
71. 13.532 years