Math 101

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Math 101

• Mathematics for Management Students

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Exponential and Logarithmic Functions

• Lectures15-16

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Lecture 16

• Logarithmic Functions

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Logarithmic Functions

• The Logarithm Function loga x is defined as
• loga x y if and only if ay x.
• a is called the base of the logarithm.

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• In calculus the most useful base for logarithms
  is the number e

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The Natural Logarithm Function

• The natural logarithm function, denoted by ln x,
  is defined as
• ln x y if and only if ey x.
• where ln x is loge x,
• that is the base is e

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The natural logarithmic function and the natural
exponential function are inverse functions

• For Example
• ln 1 0 and e0 1
• ln e 1 and e1 e

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The graph of y lnx
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Inverse properties of Logarithms and Exponents

• For Example,
• ln e3 3
• eln 5 5

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Properties of Logarithms

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

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Example

• Use the properties of logarithms to rewrite each
  expression as a sum, difference, or multiple of
  logarithms.

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Solution
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Example

• Use the properties of logarithms to rewrite each
  expression as the logarithm of single quantity

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Solution
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Solving Exponential and Logarithmic Equations

• Solve the following equations

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Solving Exponential and Logarithmic Equations

• Solution (a)
• Moving 10 to the right side, we get,
• e0.1x 14 - 10 4
• Take ln for each side, we get
• ln e0.1x ln 4
• From the property ln ex x , we get ,
• 0.1x ln 4
• x ln 4 10 ln 4

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Solving Exponential and Logarithmic Equations

• Solution (b)
• Moving 3 to the right side, we get
• 2 ln x 4
• ln x 2
• Exponentiate each side, we get
• eln x e2
• x e2

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Derivatives of Logarithmic Functions
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Example

• Find the derivative of

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Solution