4'3 Logarithmic Functions

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4.3 Logarithmic Functions
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Definition

• Let agt0 and a?1. The logarithm to the base a is
  defined by
• logaxb if and only if abx

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Special Logarithms

• The Common Logarithm
• Let a10
• log10xb if and only if 10bx
• Notation log x
• The Natural Logarithm
• Let a e
• logexb if and only if ebx
• Notation ln x

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Graph of f(x)ln x

• Domain(0, ?)
• Range (-?,?)
• Intercept (1,0)
• Always increasing
• ln x? ? as x? ?
• ln x? -? as x? 0
• Continuous
• One-to-one

ex
ln x
(1,0)
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Inverse Properties

• 1) ln ex x
• 2) eln x x

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Properties of Logarithms

• 1) ln xy ln x ln y
• 2) ln (x/y) ln x - ln y
• 3) ln xn n ln x

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Solving Exponential and Logarithm Equations

• Simplify as much as possible
• General rule is to do the opposite
• If exponential equation, take the logarithm
• If logarithmic equation, exponentiate