3'2 Logarithmic Functions

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3.2 Logarithmic Functions

• Convert between exponential and logarithmic
  equations.
• Evaluate logarithms.
• Use basic logarithm properties.
• Graph logarithmic functions.
• Find common and natural logarithms using a
  calculator.

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Logarithmic Functions

• These functions are inverses of exponential
  functions.

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Logarithmic Function, Base b

• We define y logb x as that number y such that x
  by, where x gt 0 and b is a positive constant
  other than 1.
• We read logb x as the logarithm, base b, of x.
• The function f(x) logb x is the logarithmic
  function with base b

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The equations y logb x and x by are different
ways of expressing the same thing. The first is
the logarithmic form and the second is the
exponential form. Each can be changed to the
other.
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• Convert each of the following to an exponential
  equation.
• a) log7 343 3 73 343
  
• b) logb R 12 b12 R

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Logarithms

• Convert each of the following to a logarithmic
  equation.
• a) 25 5x b) ew 30
• log5 25 x loge 30 w

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Evaluating Logarithms

• Find each of the following logarithms.
• a) log2 16 b) log10 1000
• c) log16 4 d) log10 0.001
• a) The exponent to which we raise 2 to obtain 16
  is 4 thus log2 16 4.
• b) The exponent to which we raise 10 to obtain
  1000 is 3 thus log10 1000 3.
• c) The exponent we raise 16 to get 4 is ½, so
• log16 4 ½.
• d) We have
• The exponent to which we raise 10 to get 0.001 is
  ?3, so log10 0.001 ?3.

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Basic Logarithmic Properties Involving One

• logb 1 0 because 0 is the exponent to which b
  must be raised to get 1.
• logb b 1 because 1 is the exponent to which b
  must be raised to get b.

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Recall that the composition of inverse functions
always results in x. Applying these relations to
exponential and logarithmic functions give the
following inverse properties of logs
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Evaluate
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The graph of a logarithmic function is a
reflection of the exponential function through
the line y x, since they are inverse functions.
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(0, 1)
(1, 0)
0
bgt 1
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(0, 1)
(1, 0)
a lt 1
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Graphs of Logarithmic Functions

• Graph y log3x. Convert to exponential form x
  3y.
• 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
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Graphs of Logarithmic Functions

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

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1. The x-intercept of the graph is 1. There is
no y-intercept.
2. The y-axis is a vertical asymptote of the
graph.
3. A logarithmic function is decreasing if 0
lt b lt 1 and increasing if b gt 1.
4. The graph is smooth and continuous, with no
corners or gaps.
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The domain of a function of form is the set of
all positive real numbers (x gt 0).In general the
domain of f(x) logb g(x) consist of all x for
which g(x) gt 0.
f(x) logb x
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Find the domain
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Common logarithmsif a logarithm has a base of
10, then it is referred to as a common logarithm.
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The graph of f(x) log x is similar to the
previous log graphs.
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Use the graph of f(x) log x to graph the
following. Give the domain and asmptotes..

• f(x) log (x-2)
• f(x) log x 2
• f(x) - log x

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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 1 0 and ln e 1, for the logarithmic base
  e.

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Since e is between 2 and 3 the graph of f(x) ln
x lies between the graphs of
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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 ? ln x

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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 ? 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.