2.6 Section Exercises

Verbal

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?

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Since the functions are inverses, their graphs are mirror images about the line $\text{ }y=x.\text{ }$ So for every point $\text{ }\left(a,b\right)\text{ }$ on the graph of a logarithmic function, there is a corresponding point $\text{ }\left(b,a\right)\text{ }$ on the graph of its inverse exponential function.

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. Shifting the function right or left and reflecting the function about the y-axis will affect its domain.

5. Consider the general logarithmic function $\text{ }f\left(x\right)={\mathrm{log}}_{b}\left(x\right).\text{ }$ Why can’t $\text{ }x\text{ }$ be zero?

6. Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

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Algebraic

For the following exercises, state the domain and range of the function.

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

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

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Domain: $\text{ }\left(-\infty ,\frac{1}{2}\right);\text{ }$ Range: $\text{ }\left(-\infty ,\infty \right)$

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

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

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Domain: $\text{ }\left(-\frac{17}{4},\infty \right);\text{ }$ Range: $\text{ }\left(-\infty ,\infty \right)$

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

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

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

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Domain: $\text{ }\left(5,\infty \right);\text{ }$ Vertical asymptote: $\text{ }x=5$

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

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

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Domain: $\text{ }\left(-\frac{1}{3},\infty \right);\text{ }$ Vertical asymptote: $\text{ }x=-\frac{1}{3}$

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

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

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Domain: $\text{ }\left(-3,\infty \right);\text{ }$ Vertical asymptote: $\text{ }x=-3$

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

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

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

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Domain: $\left(\frac{3}{7},\infty \right)$ ;

19. Vertical asymptote: $x=\frac{3}{7}$ ; End behavior: as $x\to {\left(\frac{3}{7}\right)}^{+},f\left(x\right)\to -\infty$ and as $x\to \infty ,f\left(x\right)\to \infty$

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

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

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Domain: $\left(-3,\infty \right)$ ; Vertical asymptote: $x=-3$ ;

22. End behavior: as $x\to -{3}^{+}$ , $f\left(x\right)\to -\infty$ and as $x\to \infty$ , $f\left(x\right)\to \infty$

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

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

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

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Domain: $\text{ }\left(1,\infty \right);\text{ }$ Range: $\text{ }\left(-\infty ,\infty \right);\text{ }$ Vertical asymptote: $\text{ }x=1;\text{ }$ x-intercept: $\text{ }\left(\frac{5}{4},0\right);\text{ }$ y-intercept: DNE

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

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

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Domain: $\text{ }\left(-\infty ,0\right);\text{ }$ Range: $\text{ }\left(-\infty ,\infty \right);\text{ }$ Vertical asymptote: $\text{ }x=0;\text{ }$ x-intercept: $\text{ }\left(-{e}^{2},0\right);\text{ }$ y-intercept: DNE

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

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

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Domain: $\text{ }\left(0,\infty \right);\text{ }$ Range: $\text{ }\left(-\infty ,\infty \right);\text{ }$ Vertical asymptote: $\text{ }x=0;\text{ }$ x-intercept: $\text{ }\left({e}^{3},0\right);\text{ }$ y-intercept: DNE

Graphical

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

Figure 17.

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

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

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

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

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

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

Figure 18.

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

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

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

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

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

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

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

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

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

Figure 19.

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

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

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

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

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

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

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

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

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

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

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

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

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

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$\text{ }f\left(x\right)={\mathrm{log}}_{2}\left(-\left(x-1\right)\right)$

51. Use $\text{ }f\left(x\right)={\mathrm{log}}_{3}\left(x\right)\text{ }$ 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.

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

The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.

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$f\left(x\right)=3{\mathrm{log}}_{4}\left(x+2\right)$

53. Use $\text{ }f\left(x\right)={\mathrm{log}}_{5}\left(x\right)\text{ }$ 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.

Technology

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

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

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$x=2$

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

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

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$x\approx \text{2}\text{.303}$

57. $2\mathrm{ln}\left(5x+1\right)=\frac{1}{2}\mathrm{ln}\left(-5x\right)+1$

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

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$x\approx -0.472$

Extensions

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

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

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The graphs of $\text{ }f\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)\text{ }$ and $\text{ }g\left(x\right)=-{\mathrm{log}}_{2}\left(x\right)\text{ }$ appear to be the same; Conjecture: for any positive base $\text{ }b\ne 1,$ $\text{ }{\mathrm{log}}_{b}\left(x\right)=-{\mathrm{log}}_{\frac{1}{b}}\left(x\right).$

61. Prove the conjecture made in the previous exercise.

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

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Recall that the argument of a logarithmic function must be positive, so we determine where $\text{ }\frac{x+2}{x-4}>0\text{ }$ . From the graph of the function $\text{ }f\left(x\right)=\frac{x+2}{x-4},$ note that the graph lies above the x-axis on the interval $\text{ }\left(-\infty ,-2\right)\text{ }$ and again to the right of the vertical asymptote, that is $\text{ }\left(4,\infty \right).\text{ }$ Therefore, the domain is $\text{ }\left(-\infty ,-2\right)\cup \left(4,\infty \right).$

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

Exponential and Logarithmic Functions