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(x)=logb(x). 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(x)=log3(x+4)
7. h(x)=ln(21−x)
8. g(x)=log5(2x+9)−2
9. h(x)=ln(4x+17)−5
10. f(x)=log2(12−3x)−3
For the following exercises, state the domain and the vertical asymptote of the function.
11. f(x)=logb(x−5)
12. g(x)=ln(3−x)
13. f(x)=log(3x+1)
14. f(x)=3log(−x)+2
15. g(x)=−ln(3x+9)−7
For the following exercises, state the domain, vertical asymptote, and end behavior of the function.
16. f(x)=ln(2−x)
17. f(x)=log(x−73)
18. h(x)=−log(3x−4)+3
19. g(x)=ln(2x+6)−5
20. f(x)=log3(15−5x)+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(x)=log4(x−1)+1
22. f(x)=log(5x+10)+3
23. g(x)=ln(−x)−2
24. f(x)=log2(x+2)−5
25. h(x)=3ln(x)−9
For the following exercises, match each function in the graph below with the letter corresponding to its graph.
26. d(x)=log(x)
27. f(x)=ln(x)
28. g(x)=log2(x)
29. h(x)=log5(x)
30. j(x)=log25(x)
For the following exercises, match each function in the figure below with the letter corresponding to its graph.
31. f(x)=log31(x)
32. g(x)=log2(x)
33. h(x)=log43(x)
For the following exercises, sketch the graphs of each pair of functions on the same axis.
34. f(x)=log(x) and g(x)=10x
35. f(x)=log(x) and g(x)=log21(x)
36. f(x)=log4(x) and g(x)=ln(x)
37. f(x)=ex and g(x)=ln(x)
For the following exercises, match each function in the graph below with the letter corresponding to its graph.
38. f(x)=log4(−x+2)
39. g(x)=−log4(x+2)
40. h(x)=log4(x+2)
For the following exercises, sketch the graph of the indicated function.
41. f(x)=log2(x+2)
42. f(x)=2log(x)
43. f(x)=ln(−x)
44. g(x)=log(4x+16)+4
45. g(x)=log(6−3x)+1
46. h(x)=−21ln(x+1)−3
For the following exercises, write a logarithmic equation corresponding to the graph shown.
47. Use y=log2(x) as the parent function.
48. Use f(x)=log3(x) as the parent function.
49. Use f(x)=log4(x) as the parent function.
50. Use f(x)=log5(x) as the parent function.
For the following exercises, use a graphing calculator to find approximate solutions to each equation.
51. log(x−1)+2=ln(x−1)+2
52. log(2x−3)+2=−log(2x−3)+5
53. ln(x−2)=−ln(x+1)
54. 2ln(5x+1)=21ln(−5x)+1
55. 31log(1−x)=log(x+1)+31
56. Let b be any positive real number such that b=1. What must logb1 be equal to? Verify the result.
57. Explore and discuss the graphs of f(x)=log21(x) and g(x)=−log2(x). 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(x)=ln(x−4x+2)? Discuss the result.
60. Use properties of exponents to find the x-intercepts of the function f(x)=log(x2+4x+4) algebraically. Show the steps for solving, and then verify the result by graphing the function.
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