The Natural Logarithmic Function

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The Natural Logarithmic Function

• Differentiation

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Definition of the Natural Logarithmic Function

• The natural logarithmic function is defined by
• The domain of the natural logarithmic function is
  the set of all positive real numbers

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Properties of the Natural Logarithmic Function

• The domain is (0, 8) and the range is
• (- 8, 8).
• The function is continuous, increasing, and
  one-to-one.
• The graph is concave downward.

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Graph of a the Natural Logarithmic Function
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Logarithmic Properties

• If a and b are positive numbers and n is
  rational, then the following properties are true.
• 1. ln (1) 0
• 2. ln(ab) ln a ln b
• 3. ln(an) n ln a
• 4. ln (a/b) ln a ln b

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Properties of Logarithms

• Use the properties of logarithms to approximate
  ln 0.25 given that
• ln 2 0.6931 and ln 3 1.0986
• (b) ln 24
• (c) ln 1/72

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Expanding Logarithmic Expressions

• Use the properties of logarithms to expand the
  logarithmic expression

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Logarithms as a Single Quantity

• Write the expression as a logarithm of a single
  quantity
• (a) 3 ln x 2 ln y 4 ln z
• (b) 2 ln 3 - ½ln (x2 1)
• (c) ½ln (x2 1) ln (x 1) ln (x 1)

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The Number e

• The base of the natural logarithmic function is e
• e 2.71828182846 . . .

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Definition of e

• The letter e denotes the positive real number
  such that

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Evaluating Natural Logarithmic Expressions

• ln2
• ln 32
• ln 0.1

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Derivative of the Natural Logarithmic Function
In other words, the derivative of the function
over the function.
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Differentiation of Logarithmic Functions

• Find the derivative of the function
• (a) h(x) ln (2x2 1)
• (b) f(x) x ln x

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Differentiation of Logarithmic Functions
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Logarithmic Properties as Aids to Differentiation
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Logarithmic Properties as Aids to Differentiation
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More Examples

• P. 322 problems 60
• On-line Examples

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Logarithmic Differentiation
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Logarithmic Differentiation
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Logarithmic Differentiation

• P. 322 problems 87 92
• On-line Examples

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Finding the Equation of the Tangent Line

• Find an equation of the tangent line to the graph
  of f at the indicated point

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Locating Relative Extrema

• Locate any relative extrema and inflection points
  for the graph of
• Y x ln x
• Y lnx/x