171S5.3q Logarithmic Functions and Graphscfcc.edu/faculty/cmoore/171ClassNotesSp13/171S5.3q.pdf · 8 is 3, so log 8 = 3.2 d) log 9 ... Solution: Method 1 y = log 5 x is equivalent

171S5.3q Logarithmic Functions and Graphs

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CHAPTER 5: Exponential and Logarithmic Functions

5.1 Inverse Functions5.2 Exponential Functions and Graphs5.3 Logarithmic Functions and Graphs5.4 Properties of Logarithmic Functions5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest

MAT 171 Precalculus AlgebraDr. Claude Moore

Cape Fear Community College

Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link.

Graphs of Logarithmic Functions & Inverse Functions: Exponential and Logarithmic

Use this program to check logarithmic function graphs. http://cfcc.edu/mathlab/geogebra/logarithmic.html

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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.

Logarithmic FunctionsThese 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.

* Note that the curve does not touch or cross the yaxis.

1. Choose values for y.2. Compute values for x.3. Plot the points and connect them with a smooth curve.

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Example

Graph: x = 2y.

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.

* Note that the curve does NOT touch or cross the yaxis.

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If base > 1, Exponential and Logarithmic graphs go UP to the RIGHT.

The graphs of f(x) = ax and f 1(x) = loga x for a > 1 and 0 < a < 1 are shown below.

If 0 < base < 1, Exponential and Logarithmic graphs

go DOWN to the RIGHT.

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.”

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Finding Certain Logarithms ExampleFind each of the following logarithms.

a) log10 10,000 b) log10 0.01 c) log2 8d) log9 3 e) log6 1 f) log8 8Solution: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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Logarithms

loga 1 = 0 and loga a = 1, for any logarithmic base a.

A logarithm is an exponent!

Example

Convert each of the following to a logarithmic equation.a) 16 = 2xb) 10–3 = 0.001c) et = 70

ExampleFind each of the following common logarithms on a calculator. Round to four decimal places.a) log 645,778 b) log 0.0000239c) 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.

ExampleFind each of the following natural logarithms on a calculator. Round to four decimal places.a) ln 645,778b) ln 0.0000239 c) log (5)d) ln e e) ln 1

http://media.pearsoncmg.com/aw/aw_bittinger_algtrig_precalc_5/bat05_ifig_launch.html

http://cfcc.edu/mathlab/geogebra/logarithmic.html


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Changing Logarithmic BasesThe ChangeofBase Formula

For any logarithmic bases a and b, and any positive number M,

Solution:First, we let a = 10, b = 5, and M = 8. Then we substitute into the changeofbase formula:

Find log5 8 using common logarithms.

Example

Solution:Substituting e for a, 6 for b and 8 for M, we have

We can also use base e for a conversion.

Find log5 8 using natural logarithms.

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Graphs of Logarithmic Functions ExampleGraph: y = f (x) = log5 x.Solution: Method 1y = log5 x is equivalent to x = 5y. Select y and compute x.

Graph: y = f (x) = log5 x.Solution: Method 2Use a graphing calculator. First change bases.

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Example (continued)

Graph: y = f (x) = log5 x.Solution: Method 3Calculators 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).

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 xc) f (x) = |ln (x – 1)|

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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.

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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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.

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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Examplea. 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:

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(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.


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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.

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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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

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430/6. Make a handdrawn graph of each of the following. Then check your work using a graphing calculator: y = log4 x

430/8. Make a handdrawn graph of each of the following. Then check your work using a graphing calculator: f(x) = ln x

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Find the following without using a calculator:

430/10. log 3 9

430/15. log 2 (1/4)

430/24. log 10 (8/5)

430/28. log3 32

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430/36. 5 3 = 1 / 125

430/40. Q t = x

Convert to a logarithmic equation:

430/42. e t = 0.3679

430/44. e t = 4000

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430/48. log 7 = 0.845

430/46. t = log4 7

Convert to an exponential equation:

430/54. ln W 5 = t

430/50. ln 0.38 = 0.9676


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430/58. log 93,100

430/62. ln 50

Find each of the following using a calculator. Round to four decimal places:

430/66. ln 0.00037

430/68. ln 0

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430/70. Find the logarithm using common logarithm and change of base formula: log 3 20

430/74. Find the logarithm using common logarithm and change of base formula: log 5.3 1700

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430/76. Find the logarithm using natural logarithm and change of base formula: log 4 25

430/78. Find the logarithm using natural logarithm and change of base formula: log 9 100

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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

430/80. 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


Nov 88:43 PM

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

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


Nov 88:46 PM

430/84. f(x) = log 3 (x 2)

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.

430/88. f(x) = (1 / 2) ln x







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430/90. f(x) = ln (x + 1)


431/92. f(x) = (1 / 2)log (x 1) 2


Reference point (2, -2).

V.A. x = 1

(x-1) moves basic graph 1 unit to right.

-2 moves basic graph down 2 units.

(1/2) multiplies y-values by 1/2.

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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

c) Virginia Beach, Virginia: 437,994

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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%?

Nov 88:48 PM

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 4c) 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?


Documents

171S5.3q Logarithmic Functions and Graphscfcc.edu/faculty/cmoore/171ClassNotesSp13/171S5.3q.pdf · 8 is 3, so log 8 = 3.2 d) log 9 ... Solution: Method 1 y = log 5 x is equivalent