Logarithmic Functions and Graphs

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Section 4.3

• Logarithmic Functions and Graphs

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Flashback

• Consider the graph of the exponential function y
  f(x) 3x.
• Is f(x) one-to-one?
• Does f(x) have an
• inverse that is a
• function?
• Find the inverse.

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Inverse of y 3x

• f (x) 3x
• y 3x
• x 3y

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• x 3y
• Now, solve for y.
• y the power to which 3 must be raised in order
  to obtain x.

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• x 3y
• Solve for y.
• y the power to which 3 must be raised in order
  to obtain x.
• Symbolically, y log 3 x
• The logarithm, base 3, of x.

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Logarithm

• For all positive numbers a, where a 1,
• Logax is an exponent to which the base a
• must be raised to give x.

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• Logarithmic Form

• Exponential Form

Argument (always positive)
All a log is . . . is an exponent!
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Logarithmic Functions

• Logarithmic functions are inverses of exponential
  functions.
• Graph x 3y or y log 3 x
• 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 y-axis.

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Logarithmic Functions continued
Graph x 3y
y log 3 x
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Side-by-Side Comparison
f (x) 3x
f (x) log 3 x
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Comparing Exponential and Logarithmic Functions
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Logarithmic Functions

• Remember Logarithmic functions are inverses of
  exponential functions.

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Asymptotes

• Recall that the horizontal asymptote of the
  exponential function y ax is the x-axis.
• The vertical asymptote of a logarithmic function
  is the y-axis.

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Logarithms

• Convert each of the following to a logarithmic
  equation.
• a) 25 5x b) ew 30

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Example

• Convert each of the following to an exponential
  equation.
• a) log7 343 3
• log7 343 3 7 3 343
• b) logb R 12

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Finding Certain Logarithms

• Find each of the following.
• a) log2 16 b) log10 1000
• c) log16 4 d) log10 0.001

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Common Logarithm
Logarithms, base 10, are called common logarithms.

• Log button on your calculator
• is the common log

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Example

• Find each of the following common logarithms on a
  calculator.
• Round to four decimal places.
• a) log 723,456
• b) log 0.0000245
• c) log (?4)

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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 button on your calculator is the natural
log
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Example

• Find each of the following natural logarithms on
  a calculator.
• Round to four decimal places.
• a) ln 723,456
• b) ln 0.0000245
• c) ln (?4)

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Changing Logarithmic Bases

• The Change-of-Base Formula
• For any logarithmic bases a and b,
• and any positive number M,

Use change of base formula when you have a
logarithm that is not base 10 or e.
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Example

• Find log6 8 using common logarithms.
• Solution First, we let a 10, b 6, and M
  8. Then we substitute into the change-of-base
  formula

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Example

• We can also use base e for a conversion.
• Find log6 8 using natural logarithms.
• Solution Substituting e for a, 6 for b and 8
  for M, we have

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Properties of Logarithms
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Graphs of Logarithmic Functions

• Graph y f(x) log6 x.
• Select y.
• Compute x.

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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 ? 2)
• b) f(x) 2 ? ¼ ln x
• c) f(x) ln (x 1)

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Graph f(x) ln (x ? 2)

• The graph is a shift 2 units right.
• The domain is the set of all real numbers greater
  than 2.
• The line x 2 is the vertical asymptote.

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Graph f(x) 2 ? ¼ ln x

• The graph is a vertical shrinking, followed by a
  reflection across the x-axis, and then a
  translation up 2 units.
• The domain is the set of all positive real
  numbers.
• The y-axis is the vertical asymptote.

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Graph f(x) ln (x 1)

• The graph is a translation 1 unit to the left.
• Then the absolute value has the effect of
  reflecting negative outputs across the x-axis.
• The domain is the set of all real numbers greater
  than ?1.
• The line x ?1 is the vertical asymptote.

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Application Walking Speed

• 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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Application Walking Speed continued

• The population of Philadelphia, Pennsylvania, is
• 1,517,600.
• Find the average walking speed of people living
  in
• Philadelphia.
• Since 1,517,600 1517.6 thousand,
• we substitute 1517.6 for P, since P is in
  thousands
• w(1517.6) 0.37 ln 1517.6 0.05
• ? 2.8 ft/sec.
• The average walking speed of people living in
  Philadelphia is about 2.8 ft/sec.