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 [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.

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