171S5.3 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
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.
November 09, 2010
5.3 Logarithmic Functions and Graphs
• 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.
Example
Graph: x = 2y.
To Graph: x = 2y.
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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171S5.3 Logarithmic Functions and Graphs
Example (continued)
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.
Finding Certain Logarithms ­ Example
Find each of the following logarithms.
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.
November 09, 2010
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.”
Example (continued)
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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171S5.3 Logarithmic Functions and Graphs
Logarithms
loga 1 = 0 and loga a = 1, for any logarithmic base a.
November 09, 2010
Example
Convert each of the following to a logarithmic equation.
a) 16 = 2x
b) 10–3 = 0.001
c) et = 70
A logarithm is an exponent!
Example
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)
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.
(b) log 0.0000239 ≈ ­4.6216
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171S5.3 Logarithmic Functions and Graphs
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
November 09, 2010
Changing Logarithmic Bases
The Change­of­Base Formula
For any logarithmic bases a and b, and any positive number M,
(a) ln 645778 ≈ 13.3782 (b) ln 0.0000239 ≈ ­10.6416 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.
Find log5 8 using natural logarithms.
Solution:
Substituting e for a, 6 for b and 8 for M, we have
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171S5.3 Logarithmic Functions and Graphs
November 09, 2010
Graphs of Logarithmic Functions ­ Example
Graph: y = f (x) = log5 x.
Example (continued)
Solution: Method 1
y = log5 x is equivalent to x = 5y. Select y and compute x.
Graph: y = f (x) = log5 x.
Example (continued)
Graph: y = f (x) = log5 x.
Solution: Method 3
Calculators which graph inverses automatically.
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).
Solution: Method 2
Use a graphing calculator. First change bases.
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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171S5.3 Logarithmic Functions and Graphs
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.
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 09, 2010
Example (continued)
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.
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
w(P) = 0.37 ln P + 0.05.
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171S5.3 Logarithmic Functions and Graphs
November 09, 2010
Example (continued)
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.
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.
Example (continued)
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.
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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171S5.3 Logarithmic Functions and Graphs
423/4. Make a hand­drawn graph of each of the following. Then check your work using a graphing calculator: x = (4/3)y
423/15. Find the following without using a calculator: log 2 (1/4)
November 09, 2010
423/10. Find the following without using a calculator: log 3 9
423/24. Find the following without using a calculator: log 10 (8/5)
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171S5.3 Logarithmic Functions and Graphs
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423/40. Convert to a logarithmic equation: Q t = x
423/50. Convert to an exponential equation: ln 0.38 = ­0.9676
ln 0.38 = ­0.9676 is equivalent to 0.38 = e ­0.9676
423/44. Convert to a logarithmic equation: e ­t = 4000
423/54. Convert to an exponential equation: ln W 5 = t
ln W 5 = t is equivalent to W 5 = e t
423/62. Find each of the following using a calculator. Round to four decimal places: ln 50
ln 50 = 3.9120.
423/74. Find the logarithm using common logarithm and change of base formula: log 5.3 1700
Check: e 3.9120 ≈ 50.
423/68. Find each of the following using a calculator. Round to four decimal places: ln 0
Zero is not in the domain of logarithm function. So, ln (0) does not exist.
Remember that logarithm (positive quantity).
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171S5.3 Logarithmic Functions and Graphs
423/76. Find the logarithm using natural logarithm and change of base formula: log 4 25
log 4 25 = (log 25) / (log 4) = 2.3219
November 09, 2010
423/80. 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
Let Y1 = log 4 x = (log x) / (log 4) and Y2 = 4 x and find where Y1 = Y2.
Y2
Y1
424/84. 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. f(x) = log 3 (x ­ 2)
f(x) = log 3 (x ­ 2) is the graph of y = log 3 x shifted 2 units to the right because x ­ 2 = 0 yields x = 2 which represents the vertical asymptote. In order to graph with the calculator, we must use the change of base formula as follows:
424/94. 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
(a) H+ = 1.6 x 10­4 So the pH = ­log(1.6 x 10­4) ≈ 3.796
f(x) = log 3 (x ­ 2) = (log (x ­ 2)) / (log 3)
(b) H+ = 6.3 x 10­7 So the pH = ­log(6.3 x 10­7) ≈ 6.201
x­intercept moves from (1, 0) to (3,0).
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