4.4e: Exercises - Graphs of Logarithmic Functions

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A: Concepts

Exercise $\PageIndex{A}$: Concepts

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 $f(x)=\log _b(x)$Why can’t $x$ be zero?

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

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

6) $d(x)=\log (x)$

7) $f(x)=\ln (x)$

8) $g(x)=\log_2 (x)$

9) $h(x)=\log_5 (x)$

10) $j(x)=\log_{25} (x)$

12) $g(x)=\log_2 (x)$

13) $f(x)=\log_{\frac{1}{3}} (x)$

14) $h(x)=\log_{\frac{3}{4}} (x)$

Answers to odd exercises:

1. Since the functions are inverses, their graphs are mirror images about the line $y-x$So for every point $(a,b)$ on the graph of a logarithmic function, there is a corresponding point $(b,a)$ on the graph of its inverse exponential function.

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

5. No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

7. $B$, 9. $C$, 13. $B$

B: Find Domain and Range

Exercise $\PageIndex{B}$: Find Domain and Range

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

16. $f(x)=\log _3(x+4)$

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

18. $g(x)=\log _5(2x+9)-2$

19. $h(x)=\ln (4x+17)-5$

20. $f(x)=\log _2 (12-3x)-3$

$ \bigstar $ State the domain of each function

21. $f(x)=\ln \left (\dfrac{x+2}{x-4} \right )$

22.1 $f(x)=\log _2 \left ( (x^2-1)(x+1) \right )$

22.2 $f(x)=\log _3 \left ( (x^2-9)(x+4) \right )$

22.3 $f(x)=\log _3 \left ( \dfrac{(x+1)^2}{(x-3)} \right )$

22.4 $f(x)=\log _2 \left ( \dfrac{x}{(x-2)^2} \right )$

Answers to odd exercises:

17. Domain: $\left ( -\infty , \dfrac{1}{2} \right )$; Range: $(-\infty , \infty )$

19. Domain: $\left ( -\dfrac{17}{4}, \infty \right )$; Range: $(-\infty , \infty )$

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

C: Find Asymptote

Exercise $\PageIndex{C}$: Find Asymptote

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

23. $f(x)=\log _b (x-5)$

24. $g(x)=\ln (3-x)$

25. $f(x)=\log (3x+1)$

26. $f(x)=3\log (-x)+2$

27. $g(x)=-\ln (3x+9)-7$

28. $f\left(x\right)=\log \left(x+2\right)$

29. $f\left(x\right)=\log \left(x-5\right)$

31. $f\left(x\right)=\ln \left(3-x\right)$

32. $f\left(x\right)=\ln \left(5-x\right)$

33. $f\left(x\right)=\log \left(3x+1\right)$

34. $f\left(x\right)=\log \left(2x+5\right) -3$

34.1. $f\left(x\right)=2\log \left(-x\right)+1$

35. $f\left(x\right)=3\log \left(-x\right)+2$

Answers to odd exercises:

23. Domain: $(5, \infty )$; V. A. $x=5$

25. Domain: $\left ( -\frac{1}{3}, \infty \right )$; V. A. $x=-\frac{1}{3}$

27. Domain: $(-3, \infty )$; V. A. $x=-3$

29. Domain: $x > 5$ V. A. @ $x = 5$

31. Domain: $x < 3$ V. A. @ $x = 3$

33. Domain: $x > -\frac{1}{3}$ V. A. @ $x = -\frac{1}{3}$

35. Domain: $x < 0$ V. A. @ $x = 0$

D: State End Behaviour

Exercise $\PageIndex{D}$: State End Behaviour

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

36. $f(x)=\ln (2-x)$

37. $f(x)=\log \left ( x-\dfrac{3}{7} \right )$

38. $h(x)=-\log (3x-4)+3$

39. $g(x)=\ln (2x+6)-5$

40. $f(x)=\log_3 (15-5x)+6$

Answers to odd exercises:

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

39. Domain: $\left ( -3,\infty \right )$, Vertical asymptote: $x=-3$, End behavior: as $x\rightarrow -3^+$, $f(x)\rightarrow -\infty$ and as $x\rightarrow \infty ,f(x)\rightarrow \infty$

E: Find Intercepts

Exercise $\PageIndex{E}$: Find Intercepts

$ \bigstar $ For the following exercises, state (a) the domain and range, and (b) $x$- and $y$-intercepts, if they exist or write NONE.

41. $h(x)=\log_4 (x-1)+1$

42. $f(x)=\log (5x+10)+3$

43. $g(x)=\ln (-x)-2$

44. $f(x)=\log_2 (x+2)-5$

45. $h(x)=3\ln (x)-9$

46. $f(x)=\log \left ( x^2+4x+4 \right )$

Answers to odd exercises:

41. Domain: $\left (1,\infty \right )$

Range: $-\infty , \infty $

Vertical asymptote: $x=1$

$x$-intercept: $\left ( \dfrac{5}{4},0\right )$

$y$-intercept: NONE

43. Domain: $\left (-\infty ,0 \right )$

Range: $-\infty , \infty $

Vertical asymptote: $x=0$

$x$-intercept: $\left ( -e^2,0 \right )$

$y$-intercept: NONE

45. Domain: $\left (0,\infty \right )$

Range: $-\infty , \infty $

Vertical asymptote: $x=0$

$x$-intercept: $\left ( e^3,0 \right )$

$y$-intercept: NONE

F: Graph Basic Logarithmic Functions

Exercise $\PageIndex{F}$: Graph Basic Logarithmic Functions

$ \bigstar $ In the following exercises, graph each logarithmic function (or functions).

47. $y=\log _{4} (x)$

48. $y=\log _{2} (x)$

49. $y=\log _{7} (x)$

50. $y=\log _{6} (x)$

51. $y=\log _{2.5} (x)$

52. $y=\log _{1.5} (x)$

53. $y=\log _{\frac{1}{5}} (x)$

54. $y=\log _{\frac{1}{3}} (x)$

55. $y=\log _{0.6} (x)$

56. $y=\log _{0.4} (x)$

57. $f(x)=\log _{1 / 2} (x)$

58. $f(x)=\log (x),$ and $ g(x)=\ln (x)$

Answers to odd exercises:

47. $This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).$ $y=\log _{4} x$	49. $This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).$ $y=\log _{7} x$	51. $This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).$ $y=\log _{2.5} x$
53. $This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).$ $y=\log _{\frac{1}{5}} x$	55. $This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).$ $y=\log _{0.6} x$	57. $7597f544c3fe5e53622ecc55094323ec.png$ $f(x)=\log _{1 / 2} x$

G: Graph Vertical and Horizontal Shifts of Basic Log Functions

Exercise $\PageIndex{G}$: Graph Vertical and Horizontal Shifts of Basic Log Functions

$ \bigstar $ In the following exercises, graph each function using transformations. State the parent function and the transformations needed to be made on the parent function in order to obtain the graph of the translated function.

62. $f(x)=\log (x)−1$

63. $f(x)=\log _{2} (x)-2$

64. $f(x)=\log _{3} (x)+3$

65. $f(x)=\log (x)+5$

66. $f(x)=\log _{1 / 3} (x)+2$

67. $f(x)=3+\ln(x)$

68. $f(x)=\log _{3}(x-2)$

69. $f(x)=\log _2(x+2)$

70. $ f(x)=\log _{3}(x+4)$

71. $f(x)=\log _{2}(x+1)$

72. $h(x)=\log _4(x+2)$

73. $f(x)=\log _{1 / 2}(x-2)$

74. $f(x)=\ln(x−1)$

75. $f(x)=\ln (x-3)$

76. $f(x)=\ln (x+5)$

77. $f(x)=\ln(x+1)$

78. $f(x)=\log (x-5)+10$

79. $f(x)=\log _{2}(x-2)+4$

80. $f(x)=\log _{3}(x+1)-2$

81. $f(x)=\log (x+4)-8$

82. $f(x)=\log _{1 / 3}(x+1)-1$

83. $f(x)=\ln (x-2)+4$

Answers to odd exercises:

63. $ y = \log_2(x) $. Down 2 $b4fc45e07f9e5dd402f7feefcfe7337f.png$ $f(x)=\log _{2} x-2$	65. $ y = \log(x) $. Up 5 $543e3bd9fd14b3ffcc942ca04645222d.png$ $f(x)=\log x+5$	67. $ y = \ln(x) $. Up 3 $CNX_Calc_Figure_01_05_214.jpeg$ $f(x)=3+\ln(x)$	69. $ y = \log_2(x) $. Left 2 $Screen Shot 2019-10-04 at 2.48.16 PM.png$ $f\left(x\right)=\log _{2} (x+2)$
71. $ y = \log_2(x) $. Left 1 $262d8241e3eee557e9ace79d00ed26cf.png$ $f(x)=\log _{2}(x+1)$	73. $ y = \log _{1 / 2}(x) $. Right 2 $330e786d26a069f20f68a6f117d8cb76.png$ $f(x)=\log _{1 / 2}(x-2)$	75. $ y = \ln(x) $. Right 3 $2a26f9ebd76f89b53eb7eca1ff058ebb.png$ $f(x)=\ln (x-3)$	77. $ y = \ln(x) $. Left 1 $An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line “x = -1”. There y intercept and the x intercept are both at the origin.$ $f(x)=ln(x+1)$
79. $ y = \log_2(x) $. Right 2, up 4 $a0fed4182de1096ba09a6929a1d4bc48.png$ $f(x)=\log _{2}(x-2)+4$	81. $ y = \log(x) $. Left 4, down 8 $fa0a4c29ef4db9b2c0fcec84ba853693.png$ $f(x)=\log (x+4)-8$	83. $ y = \ln(x) $. Right 2, up 4 $59c3c813f519710df16615aa95137fe2.png$ $f(x)=\ln (x-2)+4$

H: Graph Reflections and Transformations of Log Functions

Exercise $\PageIndex{H}$: Graph Reflections and Transformations of Log Functions

84. $f(x)=2-\log _{3}(-x)$

85. $f(x)=\log _{2}(-x)+1$

86. $f(x)=-\log _{3}(x+3)$

87. $f(x)=-\log _{2} (x)+1$

88. $f(x)=1-\log _4(-x)$

89. $f(x)=2+\log _4 (x)$

90. $f(x)=1+\log _{1 / 2}(-x)$

91. $f(x)=1-\log _{1 / 3}(x-2)$

92. $f(x)=1−\ln(x)$

93. $f(x)=2-\ln (x)$

94. $f(x)=-\ln (x-1)$

95. $f(x)=\ln(−x)$

96. $f(x)=-\log (x-1)+2$

97. $f(x)=-\log (x+2)$

98. $f(x)=2\log _3(5-x)-1$

99. $f(x)=\log (6-3x)+1$

100. $f(x)=\log _2(4x+16)+4$

102. $f(x)=4\log _3 (x-2)$ -5

103. $f(x)=2\log (x)$

104. $h(x)=-4\log _2 (x+1)-3$

Answers to odd exercises:

85. $f(x)=\log_2(x)$. Reflect over y-axis, up 1 $6703b4dd22d73a774810ac3c720c8c71.png$ $f(x)=\log _{2}(-x)+1$	87. $f(x)=\log_2(x)$. Reflect over x-axis, up 1 $a3b4330ad8cdda9afa1404c5d49266dd.png$ $f(x)=-\log _{2} x+1$	89. $f(x)=\log _4(x)$. Up 2 $17406e7b30af7957887eac740aa1aee9.png$ $f(x)=2+\log _4 x$
91. $f(x)=\log_{1 / 3}(x)$. Right 2, reflect over x-axis, up 1 $c474ed2038fe57d9f1300571e35add93.png$ $f(x)=1-\log _{1 / 3}(x-2)$	93. $f(x)=\ln(x)$. Reflect over x-axis, up 2 $28a1ac3541e46e8a2bc2475a5abc7ca6.png$ $f(x)=2-\ln x$	95. $f(x)=\ln(x)$. Reflect over y-axis $CNX_Calc_Figure_01_05_216.jpeg$ $f(x)=\ln(−x)$
97. $f(x)=\log(x)$. Left 2, reflect over x-axis $85eaece20f3696811d2e26ef2557940e.png$ $f(x)=-\log (x+2)$	99. $f(x)=\log(x)$. Left 6, reflect over y-axis, horizontally shrunk by a factor of 1/3, up 1 $Ex 4.4.43.png$ $f(x)=\log (6-3x)+1$	103. $f(x)=\log(x)$. Vertically stretched by a factor of 2 $Screen Shot 2019-10-04 at 2.47.40 PM.png$ $f\left(x\right)=2\log \left(x\right)$

I: Find the Equation for the Graph of a Logarithmic Function (no stretching)

Exercise $\PageIndex{I}$: Find the Equation for the Graph of a Logarithmic Function

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

105. $4.4e log4 L3 U1 graph.png$

106. $4.4e log3 L4 d2 graph.png$

107. ($(7,-3)$ is on the graph). $4.4e log5 r2 d4 graph.png$

108. $4.4e log2 R5 U3 graph.png$

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

109.

$Screen Shot 2019-08-13 at 5.37.13 PM.png$

110.

$Screen Shot 2019-08-13 at 5.40.03 PM.png$

111.

$Screen Shot 2019-08-13 at 5.45.38 PM.png$

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

113.

$4.4e log2 L1 Ry graph.png$

114.

$4.4e #114.png$

Answers to odd exercises:

105. $f(x)=\log _4(x+3)+1$
107. $f(x)=\log _5(x-2)-4$

109. $y = -\log_2 (x) $
111. $y = -\log_2 (x+2) $

113. $f(x)=\log _2(-(x-1))$

I: Find the Equation for the Graph of a Logarithmic Function (vertical stretching, no vertical shifting)

Exercise $\PageIndex{I}$: Find the Equation for the Graph of a Logarithmic Function

$ \bigstar $ Write a logarithmic equation corresponding to the graph shown.

120.

$Screen Shot 2019-08-13 at 5.44.43 PM.png$

121.

$Screen Shot 2019-08-13 at 5.42.46 PM.png$

122.

$Screen Shot 2019-08-13 at 5.41.17 PM.png$

123.

$CNX_PreCalc_Figure_04_04_216.jpg$

124.

$屏幕快照 2019-06-26 下午6.29.26.png$

125.

$屏幕快照 2019-06-26 下午6.29.08.png$

$ \bigstar $ Write a logarithmic equation corresponding to the graph shown.

129. $(-1,-3)$ is on the graph

$屏幕快照 2019-06-26 下午6.28.25.png$

130.

$屏幕快照 2019-06-26 下午6.28.40.png$

131. $(-6,2)$ is on the graph

$4.4e #131 2log5 (-x-1).png$

132.

$屏幕快照 2019-06-26 下午6.28.07.png$

$ \bigstar $ Write a logarithmic equation corresponding to the graph shown.

135.

$屏幕快照 2019-06-26 下午6.29.43.png$

136.

$屏幕快照 2019-06-26 下午6.30.01.png$

Answers to odd exercises:

121. $y = 4 \log_5 (x+2) $

123. $f(x)=3\log _4(x+2)$

125. $y = 3 \log_4 (x + 2)$

129. $y = -3\log _3(x+4)$

131. $y = 2\log _5(-x-1)$

135. $y = -2\log_5 (-(x - 5))$

$ \star $

47. $This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).$ \(y=\log _{4} x\)	49. $This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).$ \(y=\log _{7} x\)	51. $This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).$ \(y=\log _{2.5} x\)
53. $This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).$ \(y=\log _{\frac{1}{5}} x\)	55. $This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).$ \(y=\log _{0.6} x\)	57. $7597f544c3fe5e53622ecc55094323ec.png$ \(f(x)=\log _{1 / 2} x\)

63. \( y = \log_2(x) \). Down 2 $b4fc45e07f9e5dd402f7feefcfe7337f.png$ \(f(x)=\log _{2} x-2\)	65. \( y = \log(x) \). Up 5 $543e3bd9fd14b3ffcc942ca04645222d.png$ \(f(x)=\log x+5\)	67. \( y = \ln(x) \). Up 3 $CNX_Calc_Figure_01_05_214.jpeg$ \(f(x)=3+\ln(x)\)	69. \( y = \log_2(x) \). Left 2 $Screen Shot 2019-10-04 at 2.48.16 PM.png$ \(f\left(x\right)=\log _{2} (x+2)\)
71. \( y = \log_2(x) \). Left 1 $262d8241e3eee557e9ace79d00ed26cf.png$ \(f(x)=\log _{2}(x+1)\)	73. \( y = \log _{1 / 2}(x) \). Right 2 $330e786d26a069f20f68a6f117d8cb76.png$ \(f(x)=\log _{1 / 2}(x-2)\)	75. \( y = \ln(x) \). Right 3 $2a26f9ebd76f89b53eb7eca1ff058ebb.png$ \(f(x)=\ln (x-3)\)	77. \( y = \ln(x) \). Left 1 $An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line “x = -1”. There y intercept and the x intercept are both at the origin.$ \(f(x)=ln(x+1)\)
79. \( y = \log_2(x) \). Right 2, up 4 $a0fed4182de1096ba09a6929a1d4bc48.png$ \(f(x)=\log _{2}(x-2)+4\)	81. \( y = \log(x) \). Left 4, down 8 $fa0a4c29ef4db9b2c0fcec84ba853693.png$ \(f(x)=\log (x+4)-8\)	83. \( y = \ln(x) \). Right 2, up 4 $59c3c813f519710df16615aa95137fe2.png$ \(f(x)=\ln (x-2)+4\)

85. \(f(x)=\log_2(x)\). Reflect over y-axis, up 1 $6703b4dd22d73a774810ac3c720c8c71.png$ \(f(x)=\log _{2}(-x)+1\)	87. \(f(x)=\log_2(x)\). Reflect over x-axis, up 1 $a3b4330ad8cdda9afa1404c5d49266dd.png$ \(f(x)=-\log _{2} x+1\)	89. \(f(x)=\log _4(x)\). Up 2 $17406e7b30af7957887eac740aa1aee9.png$ \(f(x)=2+\log _4 x\)
91. \(f(x)=\log_{1 / 3}(x)\). Right 2, reflect over x-axis, up 1 $c474ed2038fe57d9f1300571e35add93.png$ \(f(x)=1-\log _{1 / 3}(x-2)\)	93. \(f(x)=\ln(x)\). Reflect over x-axis, up 2 $28a1ac3541e46e8a2bc2475a5abc7ca6.png$ \(f(x)=2-\ln x\)	95. \(f(x)=\ln(x)\). Reflect over y-axis $CNX_Calc_Figure_01_05_216.jpeg$ \(f(x)=\ln(−x)\)
97. \(f(x)=\log(x)\). Left 2, reflect over x-axis $85eaece20f3696811d2e26ef2557940e.png$ \(f(x)=-\log (x+2)\)	99. \(f(x)=\log(x)\). Left 6, reflect over y-axis, horizontally shrunk by a factor of 1/3, up 1 $Ex 4.4.43.png$ \(f(x)=\log (6-3x)+1\)	103. \(f(x)=\log(x)\). Vertically stretched by a factor of 2 $Screen Shot 2019-10-04 at 2.47.40 PM.png$ \(f\left(x\right)=2\log \left(x\right)\)