1. The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?
2. What type(s) of translation(s), if any, affect the range of a logarithmic function?
3. What type(s) of translation(s), if any, affect the domain of a logarithmic function?
4. Consider the general logarithmic function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]. Why can’t x be zero?
5. Does the graph of a general logarithmic function have a horizontal asymptote? Explain.
For the following exercises, state the domain and range of the function.
6. [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x+4\right)[/latex]
7. [latex]h\left(x\right)=\mathrm{ln}\left(\frac{1}{2}-x\right)[/latex]
8. [latex]g\left(x\right)={\mathrm{log}}_{5}\left(2x+9\right)-2[/latex]
9. [latex]h\left(x\right)=\mathrm{ln}\left(4x+17\right)-5[/latex]
10. [latex]f\left(x\right)={\mathrm{log}}_{2}\left(12 - 3x\right)-3[/latex]
For the following exercises, state the domain and the vertical asymptote of the function.
11. [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x - 5\right)[/latex]
12. [latex]g\left(x\right)=\mathrm{ln}\left(3-x\right)[/latex]
13. [latex]f\left(x\right)=\mathrm{log}\left(3x+1\right)[/latex]
14. [latex]f\left(x\right)=3\mathrm{log}\left(-x\right)+2[/latex]
15. [latex]g\left(x\right)=-\mathrm{ln}\left(3x+9\right)-7[/latex]
For the following exercises, state the domain, vertical asymptote, and end behavior of the function.
16. [latex]f\left(x\right)=\mathrm{ln}\left(2-x\right)[/latex]
17. [latex]f\left(x\right)=\mathrm{log}\left(x-\frac{3}{7}\right)[/latex]
18. [latex]h\left(x\right)=-\mathrm{log}\left(3x - 4\right)+3[/latex]
19. [latex]g\left(x\right)=\mathrm{ln}\left(2x+6\right)-5[/latex]
20. [latex]f\left(x\right)={\mathrm{log}}_{3}\left(15 - 5x\right)+6[/latex]
For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.
21. [latex]h\left(x\right)={\mathrm{log}}_{4}\left(x - 1\right)+1[/latex]
22. [latex]f\left(x\right)=\mathrm{log}\left(5x+10\right)+3[/latex]
23. [latex]g\left(x\right)=\mathrm{ln}\left(-x\right)-2[/latex]
24. [latex]f\left(x\right)={\mathrm{log}}_{2}\left(x+2\right)-5[/latex]
25. [latex]h\left(x\right)=3\mathrm{ln}\left(x\right)-9[/latex]
For the following exercises, match each function in the graph below with the letter corresponding to its graph.
26. [latex]d\left(x\right)=\mathrm{log}\left(x\right)[/latex]
27. [latex]f\left(x\right)=\mathrm{ln}\left(x\right)[/latex]
28. [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex]
29. [latex]h\left(x\right)={\mathrm{log}}_{5}\left(x\right)[/latex]
30. [latex]j\left(x\right)={\mathrm{log}}_{25}\left(x\right)[/latex]
For the following exercises, match each function in the figure below with the letter corresponding to its graph.
31. [latex]f\left(x\right)={\mathrm{log}}_{\frac{1}{3}}\left(x\right)[/latex]
32. [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex]
33. [latex]h\left(x\right)={\mathrm{log}}_{\frac{3}{4}}\left(x\right)[/latex]
For the following exercises, sketch the graphs of each pair of functions on the same axis.
34. [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex] and [latex]g\left(x\right)={10}^{x}[/latex]
35. [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex] and [latex]g\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)[/latex]
36. [latex]f\left(x\right)={\mathrm{log}}_{4}\left(x\right)[/latex] and [latex]g\left(x\right)=\mathrm{ln}\left(x\right)[/latex]
37. [latex]f\left(x\right)={e}^{x}[/latex] and [latex]g\left(x\right)=\mathrm{ln}\left(x\right)[/latex]
For the following exercises, match each function in the graph below with the letter corresponding to its graph.
38. [latex]f\left(x\right)={\mathrm{log}}_{4}\left(-x+2\right)[/latex]
39. [latex]g\left(x\right)=-{\mathrm{log}}_{4}\left(x+2\right)[/latex]
40. [latex]h\left(x\right)={\mathrm{log}}_{4}\left(x+2\right)[/latex]
For the following exercises, sketch the graph of the indicated function.
41. [latex]f\left(x\right)={\mathrm{log}}_{2}\left(x+2\right)[/latex]
42. [latex]f\left(x\right)=2\mathrm{log}\left(x\right)[/latex]
43. [latex]f\left(x\right)=\mathrm{ln}\left(-x\right)[/latex]
44. [latex]g\left(x\right)=\mathrm{log}\left(4x+16\right)+4[/latex]
45. [latex]g\left(x\right)=\mathrm{log}\left(6 - 3x\right)+1[/latex]
46. [latex]h\left(x\right)=-\frac{1}{2}\mathrm{ln}\left(x+1\right)-3[/latex]
For the following exercises, write a logarithmic equation corresponding to the graph shown.
47. Use [latex]y={\mathrm{log}}_{2}\left(x\right)[/latex] as the parent function.
48. Use [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)[/latex] as the parent function.
49. Use [latex]f\left(x\right)={\mathrm{log}}_{4}\left(x\right)[/latex] as the parent function.
50. Use [latex]f\left(x\right)={\mathrm{log}}_{5}\left(x\right)[/latex] as the parent function.
For the following exercises, use a graphing calculator to find approximate solutions to each equation.
51. [latex]\mathrm{log}\left(x - 1\right)+2=\mathrm{ln}\left(x - 1\right)+2[/latex]
52. [latex]\mathrm{log}\left(2x - 3\right)+2=-\mathrm{log}\left(2x - 3\right)+5[/latex]
53. [latex]\mathrm{ln}\left(x - 2\right)=-\mathrm{ln}\left(x+1\right)[/latex]
54. [latex]2\mathrm{ln}\left(5x+1\right)=\frac{1}{2}\mathrm{ln}\left(-5x\right)+1[/latex]
55. [latex]\frac{1}{3}\mathrm{log}\left(1-x\right)=\mathrm{log}\left(x+1\right)+\frac{1}{3}[/latex]
56. Let b be any positive real number such that [latex]b\ne 1[/latex]. What must [latex]{\mathrm{log}}_{b}1[/latex] be equal to? Verify the result.
57. Explore and discuss the graphs of [latex]f\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)[/latex] and [latex]g\left(x\right)=-{\mathrm{log}}_{2}\left(x\right)[/latex]. Make a conjecture based on the result.
58. Prove the conjecture made in the previous exercise.
59. What is the domain of the function [latex]f\left(x\right)=\mathrm{ln}\left(\frac{x+2}{x - 4}\right)[/latex]? Discuss the result.
60. Use properties of exponents to find the x-intercepts of the function [latex]f\left(x\right)=\mathrm{log}\left({x}^{2}+4x+4\right)[/latex] algebraically. Show the steps for solving, and then verify the result by graphing the function.