## Section Exercises

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 $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$. 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. $f\left(x\right)={\mathrm{log}}_{3}\left(x+4\right)$

7. $h\left(x\right)=\mathrm{ln}\left(\frac{1}{2}-x\right)$

8. $g\left(x\right)={\mathrm{log}}_{5}\left(2x+9\right)-2$

9. $h\left(x\right)=\mathrm{ln}\left(4x+17\right)-5$

10. $f\left(x\right)={\mathrm{log}}_{2}\left(12 - 3x\right)-3$

For the following exercises, state the domain and the vertical asymptote of the function.

11. $f\left(x\right)={\mathrm{log}}_{b}\left(x - 5\right)$

12. $g\left(x\right)=\mathrm{ln}\left(3-x\right)$

13. $f\left(x\right)=\mathrm{log}\left(3x+1\right)$

14. $f\left(x\right)=3\mathrm{log}\left(-x\right)+2$

15. $g\left(x\right)=-\mathrm{ln}\left(3x+9\right)-7$

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.

16. $f\left(x\right)=\mathrm{ln}\left(2-x\right)$

17. $f\left(x\right)=\mathrm{log}\left(x-\frac{3}{7}\right)$

18. $h\left(x\right)=-\mathrm{log}\left(3x - 4\right)+3$

19. $g\left(x\right)=\mathrm{ln}\left(2x+6\right)-5$

20. $f\left(x\right)={\mathrm{log}}_{3}\left(15 - 5x\right)+6$

For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

21. $h\left(x\right)={\mathrm{log}}_{4}\left(x - 1\right)+1$

22. $f\left(x\right)=\mathrm{log}\left(5x+10\right)+3$

23. $g\left(x\right)=\mathrm{ln}\left(-x\right)-2$

24. $f\left(x\right)={\mathrm{log}}_{2}\left(x+2\right)-5$

25. $h\left(x\right)=3\mathrm{ln}\left(x\right)-9$

For the following exercises, match each function in the graph below with the letter corresponding to its graph.

26. $d\left(x\right)=\mathrm{log}\left(x\right)$

27. $f\left(x\right)=\mathrm{ln}\left(x\right)$

28. $g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$

29. $h\left(x\right)={\mathrm{log}}_{5}\left(x\right)$

30. $j\left(x\right)={\mathrm{log}}_{25}\left(x\right)$

For the following exercises, match each function in the figure below with the letter corresponding to its graph.

31. $f\left(x\right)={\mathrm{log}}_{\frac{1}{3}}\left(x\right)$

32. $g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$

33. $h\left(x\right)={\mathrm{log}}_{\frac{3}{4}}\left(x\right)$

For the following exercises, sketch the graphs of each pair of functions on the same axis.

34. $f\left(x\right)=\mathrm{log}\left(x\right)$ and $g\left(x\right)={10}^{x}$

35. $f\left(x\right)=\mathrm{log}\left(x\right)$ and $g\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)$

36. $f\left(x\right)={\mathrm{log}}_{4}\left(x\right)$ and $g\left(x\right)=\mathrm{ln}\left(x\right)$

37. $f\left(x\right)={e}^{x}$ and $g\left(x\right)=\mathrm{ln}\left(x\right)$

For the following exercises, match each function in the graph below with the letter corresponding to its graph.

38. $f\left(x\right)={\mathrm{log}}_{4}\left(-x+2\right)$

39. $g\left(x\right)=-{\mathrm{log}}_{4}\left(x+2\right)$

40. $h\left(x\right)={\mathrm{log}}_{4}\left(x+2\right)$

For the following exercises, sketch the graph of the indicated function.

41. $f\left(x\right)={\mathrm{log}}_{2}\left(x+2\right)$

42. $f\left(x\right)=2\mathrm{log}\left(x\right)$

43. $f\left(x\right)=\mathrm{ln}\left(-x\right)$

44. $g\left(x\right)=\mathrm{log}\left(4x+16\right)+4$

45. $g\left(x\right)=\mathrm{log}\left(6 - 3x\right)+1$

46. $h\left(x\right)=-\frac{1}{2}\mathrm{ln}\left(x+1\right)-3$

For the following exercises, write a logarithmic equation corresponding to the graph shown.

47. Use $y={\mathrm{log}}_{2}\left(x\right)$ as the parent function.

48. Use $f\left(x\right)={\mathrm{log}}_{3}\left(x\right)$ as the parent function.

49. Use $f\left(x\right)={\mathrm{log}}_{4}\left(x\right)$ as the parent function.

50. Use $f\left(x\right)={\mathrm{log}}_{5}\left(x\right)$ as the parent function.

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

51. $\mathrm{log}\left(x - 1\right)+2=\mathrm{ln}\left(x - 1\right)+2$

52. $\mathrm{log}\left(2x - 3\right)+2=-\mathrm{log}\left(2x - 3\right)+5$

53. $\mathrm{ln}\left(x - 2\right)=-\mathrm{ln}\left(x+1\right)$

54. $2\mathrm{ln}\left(5x+1\right)=\frac{1}{2}\mathrm{ln}\left(-5x\right)+1$

55. $\frac{1}{3}\mathrm{log}\left(1-x\right)=\mathrm{log}\left(x+1\right)+\frac{1}{3}$

56. Let b be any positive real number such that $b\ne 1$. What must ${\mathrm{log}}_{b}1$ be equal to? Verify the result.

57. Explore and discuss the graphs of $f\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)$ and $g\left(x\right)=-{\mathrm{log}}_{2}\left(x\right)$. Make a conjecture based on the result.

58. Prove the conjecture made in the previous exercise.

59. What is the domain of the function $f\left(x\right)=\mathrm{ln}\left(\frac{x+2}{x - 4}\right)$? Discuss the result.

60. Use properties of exponents to find the x-intercepts of the function $f\left(x\right)=\mathrm{log}\left({x}^{2}+4x+4\right)$ algebraically. Show the steps for solving, and then verify the result by graphing the function.