Problem Set 24: Graphs of Logarithmic Functions

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.
Graph of five logarithmic functions.

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.
Graph of three logarithmic functions.

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.
Graph of three logarithmic functions.
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.
The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.

48. Use [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)[/latex] as the parent function.
The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.

49. Use [latex]f\left(x\right)={\mathrm{log}}_{4}\left(x\right)[/latex] as the parent function.
The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.

50. Use [latex]f\left(x\right)={\mathrm{log}}_{5}\left(x\right)[/latex] as the parent function.
The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.

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.