171S5.3p Logarithmic Functions and Graphs
November 15, 2012
5.3 Logarithmic Functions and Graphs
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions and Graphs
5.3 Logarithmic Functions and Graphs
5.4 Properties of Logarithmic Functions
5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest
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Graphs of Logarithmic Functions & Inverse Functions: Exponential and Logarithmic
• Find common logarithms and natural logarithms with and without a calculator.
• Convert between exponential and logarithmic equations.
• Change logarithmic bases.
• Graph logarithmic functions.
• Solve applied problems involving logarithmic functions.
Logarithmic Functions
These functions are inverses of exponential functions. We can draw the graph of the inverse of an exponential function by interchanging x and y.
To Graph: x = 2y.
Use this program to check logarithmic function graphs. http://cfcc.edu/mathlab/geogebra/logarithmic.html
1. Choose values for y.
2. Compute values for x.
3. Plot the points and connect them with a smooth curve.
* Note that the curve does not touch or cross the y­axis.
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Example
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Graph: x = 2y.
The graphs of f(x) = ax and f ­1(x) = loga x for a > 1 and 0 < a < 1 are shown below.
This curve looks like the graph of y = 2x reflected across the line y = x, as we would expect for an inverse. The inverse of y = 2x is x = 2y.
If base > 1, Exponential and Logarithmic graphs go UP to the RIGHT.
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If 0 < base < 1, Exponential and Logarithmic graphs go DOWN to the RIGHT.
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Finding Certain Logarithms ­ Example
Find each of the following logarithms.
Logarithmic Function, Base a
We define y = loga x as that number y such that x = a y, where x > 0 and a is a positive constant other than 1.
We read loga x as “the logarithm, base a, of x.”
a) log10 10,000 b) log10 0.01 c) log2 8
d) log9 3 e) log6 1 f) log8 8
Solution:
a) The exponent to which we raise 10 to obtain 10,000 is 4; thus log10 10,000 = 4.
b)
The exponent to which we raise 10 to get 0.01 is –2, so log10 0.01 = –2.
c) log2 8: 8 = 23. The exponent to which we raise 2 to get 8 is 3, so log2 8 = 3.
d) log9 3: 3 = 91/2. The exponent to which we raise 9 to get 3 is 1/2, so log9 3 = 1/2.
e) log6 1: 1 = 60. The exponent to which we raise 6 to get 1 is 0, so log6 1 = 0.
f) log8 8: 8 = 81. The exponent to which we raise 8 to get 8 is 4, so log8 8 = 1.
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171S5.3p Logarithmic Functions and Graphs
November 15, 2012
Logarithms
loga 1 = 0 and loga a = 1, for any logarithmic base a.
A logarithm is an exponent!
Natural Logarithms
Logarithms, base e, are called natural logarithms. The abbreviation “ln” is generally used for natural logarithms. Thus,
ln x means loge x.
ln 1 = 0 and ln e = 1, for the logarithmic base e.
Example
Example
Convert each of the following to a logarithmic equation.
a) 16 = 2x b) 10–3 = 0.001 c) et = 70
Example
Find each of the following natural logarithms on a calculator. Round to four decimal places.
a) ln 645,778 b) ln 0.0000239 c) log (­5)
d) ln e e) ln 1
Find each of the following common logarithms on a calculator. Round to four decimal places.
a) log 645,778 b) log 0.0000239 c) log (­3)
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Changing Logarithmic Bases
The Change­of­Base Formula
For any logarithmic bases a and b, and any positive number M,
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Graphs of Logarithmic Functions ­ Example
Graph: y = f (x) = log5 x.
Solution: Method 1
y = log5 x is equivalent to x = 5y. Select y and compute x.
Find log5 8 using common logarithms.
Solution:
First, we let a = 10, b = 5, and M = 8. Then we substitute into the change­of­base formula:
Example
We can also use base e for a conversion.
Graph: y = f (x) = log5 x.
Find log5 8 using natural logarithms.
Solution: Method 2
Use a graphing calculator. First change bases.
Solution:
Substituting e for a, 6 for b and 8 for M, we have
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Example (continued)
Graph: y = f (x) = log5 x.
Solution: Method 3
Calculators which graph inverses automatically.
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Example (continued)
a) f (x) = ln (x + 3) The graph is a shift 3 units left. The domain is the set of all real numbers greater than –3, (–3, ∞). The line x = –3 is the vertical asymptote.
Begin with Y1 = 5x, the graphs of both Y1 and its inverse Y2 = log 5 x will be drawn.
If calculator does not graph inverse function, graph Y2 = (log x) / (log 5) or Y2 = (ln x) / (ln 5).
Example
Graph each of the following. Describe how each graph can be obtained from the graph of y = ln x. Give the domain and the vertical asymptote of each function.
a) f (x) = ln (x + 3)
b) f (x) = 3 ­ ln x
c) f (x) = |ln (x – 1)|
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b) f (x) = 3 – ln x The graph is a vertical shrinking of y = ln x, followed by a reflection across the x­axis and a translation up 3 units. The domain is the set of all positive real numbers, (0, ∞). The y­axis is the vertical asymptote.
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171S5.3p Logarithmic Functions and Graphs
Example (continued)
c) f (x) = |ln (x – 1)|
The graph is a translation of y = ln x, right 1 unit. The effect of the absolute is to reflect the negative output across the x­axis. The domain is the set of all positive real numbers greater than 1, (1, ∞). The line x =1 is the vertical asymptote.
November 15, 2012
Example
a. The population of Hartford, Connecticut, is 124,848. Find the average walking speed of people living in Hartford.
b. The population of San Antonio, Texas, is 1,236,249. Find the average walking speed of people living in San Antonio.
c. Graph the function.
d. A sociologist computes the average walking speed in a city to the approximately 2.0 ft/sec. Use this information to estimate the population of the city.
Solution:
a. Since P is in thousands and 124,848 = 124.848 thousand, we substitute 124.848 for P:
w(124.848) = 0.37 ln 124.848 + 0.05
≈ 1.8 ft/sec.
The average walking speed of people living in Hartford is about 1.8 ft/sec.
Application
In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function
b. Substitute 1236.249 for P:
w(1236.249) = 0.37 ln 1236.249 + 0.05
≈ 2.7 ft/sec.
The average walking speed of people living in San Antonio is about 2.7 ft/sec.
w(P) = 0.37 ln P + 0.05.
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Example (continued)
c. Graph with a viewing window [0, 600, 0, 4] because inputs are very large and outputs are very small by comparison.
d. To find the population for which the walking speed is 2.0 ft/sec, we substitute 2.0 for w(P), 2.0 = 0.37 ln P + 0.05, and solve for P.
Use the Intersect method.
Graph Y1 = 0.37 ln x + 0.05 and Y2 = 2.
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430/2. Make a hand­drawn graph of each of the following. Then check your work using a graphing calculator: x = 4y
430/4. Make a hand­drawn graph of each of the following. Then check your work using a graphing calculator: x = (4/3)y
In a city with an average walking speed of 2.0 ft/sec, the population is about 194.5 thousand or 194,500.
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Find the following without using a calculator: 430/6. Make a hand­drawn graph of each of the following. Then check your work using a graphing calculator: y = log4 x 430/10. log 3 9 430/15. log 2 (1/4) Find the following without using a calculator: 430/24. log 10 (8/5)
430/8. Make a hand­drawn graph of each of the following. Then check your work using a graphing calculator: f(x) = ln x Find the following without using a calculator: 430/28. log3 3­2
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171S5.3p Logarithmic Functions and Graphs
Convert to a logarithmic equation:
430/36. 5 ­3 = 1 / 125
430/40. Q t = x
November 15, 2012
Convert to an exponential equation:
430/46. t = log4 7
430/48. log 7 = 0.845
Convert to a logarithmic equation:
430/42. e ­t = 0.3679
Convert to an exponential equation:
430/50. ln 0.38 = ­0.9676
Convert to a logarithmic equation:
430/44. e ­t = 4000
Convert to an exponential equation:
430/54. ln W 5 = t
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Find each of the following using a calculator. Round to four decimal places: 430/58. log 93,100
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430/70. Find the logarithm using common logarithm and change of base formula: log 3 20
430/62. ln 50
430/66. ln 0.00037
430/74. Find the logarithm using common logarithm and change of base formula: log 5.3 1700
430/68. ln 0
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430/76. Find the logarithm using natural logarithm and change of base formula: log 4 25
430/79. Graph the function and its inverse using the same set of axes. Use any method: f(x) = 3 x , f ­1(x) = log 3 x
430/78. Find the logarithm using natural logarithm and change of base formula: log 9 100
430/80. Graph the function and its inverse using the same set of axes. Use any method: f(x) = log 4 x , f ­1(x) = 4 x
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171S5.3p Logarithmic Functions and Graphs
430/81. Graph the function and its inverse using the same set of axes. Use any method: f(x) = log x , f ­1(x) = 10 x
November 15, 2012
For each of the following functions, briefly describe how the graph can be obtained from the graph of a basic logarithmic function. Then graph the function using a graphing calculator. Give the domain and the vertical asymptote of each function.
430/84. f(x) = log 3 (x ­ 2)
430/82. Graph the function and its inverse using the same set of axes. Use any method: f(x) = e x , f ­1(x) = ln x
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For each of the following functions, briefly describe how the graph can be obtained from the graph of a basic logarithmic function. Then graph the function using a graphing calculator. Give the domain and the vertical asymptote of each function.
430/90. f(x) = ln (x + 1)
430/88. f(x) = (1 / 2) ln x
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431/95. Walking Speed. Refer to Example 12. Various cities and their populations are given below. Find the average walking speed in each city. The average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function w(P) = 0.37 ln P + 0.05 (Source: International Journal of Psychology).
a) Seattle, Washington: 608,660 b) Los Angeles, California: 3,792,621 431/91. f(x) = (1 / 2)log (x ­ 1) ­ 2
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c) Virginia Beach, Virginia: 437,994 Nov 8­8:48 PM
431/96. Forgetting. Students in a computer science class took a final exam and then took equivalent forms of the exam at monthly intervals thereafter. The average score S(t), as a percent, after t months was found to be given by the function S(t) = 78 ­ 15 log (t + 1), t ≥ 0. a) What was the average score when the students initially took the test, t = 0? b) What was the average score after 4 months? after 24 months? c) Graph the function. d) After what time t was the average score 50%?
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171S5.3p Logarithmic Functions and Graphs
November 15, 2012
431/98. pH of Substances in Chemistry. In chemistry, the pH of a substance is defined as pH = ­ log [H+], where H+ is the hydrogen ion concentration, in moles per liter. Find the pH of each substance. a) Pineapple juice; Hydrogen Ion Concentration 1.6 x 10 ­4
c) Mouthwash; Hydrogen Ion Concentration 6.3 x 10 ­7
431/100. Advertising. A model for advertising response is given by the function N(a) = 1000 + 200 ln a, a ≥ 1, where N(a) is the number of units sold when a is the amount spent on advertising, in thousands of dollars.
a) How many units were sold after spending $1000 (a = 1) on advertising? b) How many units were sold after spending $5000?
c) Graph the function. d) How much would have to be spent in order to sell 2000 units? Nov 8­8:48 PM
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