Logarithmic Functions

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Logarithmic Functions
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Definition of a Logarithmic Function

• For x gt 0 and b gt 0, b 1,
• y logb x is equivalent to by x.
• The function f (x) logb x is the logarithmic
  function with base b.

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Location of Base and Exponent in Exponential and
Logarithmic Forms
Logarithmic form y logb x Exponential
Form by x.
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Text Example
Write each equation in its equivalent exponential
form. a. 2 log5 x b. 3 logb 64 c. log3 7
y
Solution With the fact that y logb x means by
x,
c. log3 7 y or y log3 7 means 3y 7.
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Text Example
Evaluate a. log2 16 b. log3 9 c. log25
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Solution
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Basic Logarithmic Properties Involving One

• Logb b 1 because 1 is the exponent to which b
  must be raised to obtain b. (b1 b).
• Logb 1 0 because 0 is the exponent to which b
  must be raised to obtain 1. (b0 1).

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Inverse Properties of Logarithms

• For x gt 0 and b ? 1,
• logb bx x The logarithm with base b of b
  raised to a power equals that power.
• b logb x x b raised to the logarithm with base
  b of a number equals that number.

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Properties of Common Logarithms

• General Properties Common Logarithms
• 1. logb 1 0 1. log 1 0
• 2. logb b 1 2. log 10 1
• 3. logb bx 0 3. log 10x x
• 4. b logb x x 4. 10 log x x

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Examples of Logarithmic Properties

• log 4 4 1
• log 8 1 0
• 3 log 3 6 6
• log 5 5 3 3
• 2 log 2 7 7

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Properties of Natural Logarithms

• General Properties Natural Logarithms
• 1. logb 1 0 1. ln 1 0
• 2. logb b 1 2. ln e 1
• 3. logb bx 0 3. ln ex x
• 4. b logb x x 4. e ln x x

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Examples of Natural Logarithmic Properties

• e log e 6 e ln 6 6
• log e e 3 3

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Problems

• Use the inverse properties to simplify

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Characteristics of the Graphs of Logarithmic
Functions of the Form f(x) logbx

• The x-intercept is 1. There is no y-intercept.
• The y-axis is a vertical asymptote. (x 0)
• If 0 lt b lt 1, the function is decreasing. If b gt
  1, the function is increasing.
• The graph is smooth and continuous. It has no
  sharp corners or edges.

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Text Example
Graph f (x) 2x and g(x) log2 x in the same
rectangular coordinate system.
Solution We first set up a table of coordinates
for f (x) 2x. Reversing these coordinates gives
the coordinates for the inverse function, g(x)
log2 x.
Reverse coordinates.
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Text Example
Graph f (x) 2x and g(x) log2 x in the same
rectangular coordinate system.
Solution
We now sketch the basic exponential graph. The
graph of the inverse (logarithmic) can also be
drawn by reflecting the graph of f (x) 2x over
the line y x.
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Examples
Graph using transformations.
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Domain of Logarithmic Functions

• Because the logarithmic function is the inverse
  of the exponential function, its domain and range
  are the reversed.
• The domain is x x gt 0 and the range will be
  all real numbers.
• For variations of the basic graph, say
  the domain will consist of all x for which x c
  gt 0.
• Find the domain of the following
• 1.
• 2.
• 3.

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Sample Problems

• Find the domain of