Properties of Logarithms

1
Properties of Logarithms
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The Product Rule

• Let b, M, and N be positive real numbers with b
  ? 1.
• logb (MN) logb M logb N
• The logarithm of a product is the sum of the
  logarithms.
• For example, we can use the product rule to
  expand ln (4x)
• ln (4x) ln 4 ln x.

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The Quotient Rule

• Let b, M and N be positive real numbers with b ?
  1.
• The logarithm of a quotient is the difference of
  the logarithms.

4
The Power Rule

• Let b, M, and N be positive real numbers with b
  1, and let p be any real number.
• log b M p p log b M
• The logarithm of a number with an exponent is the
  product of the exponent and the logarithm of that
  number.

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Text Example
Write as a single logarithm a. log4 2 log4 32
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Problems

• Write the following as single logarithms

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The Change-of-Base Property

• For any logarithmic bases a and b, and any
  positive number M,
• The logarithm of M with base b is equal to the
  logarithm of M with any new base divided by the
  logarithm of b with that new base.

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Example Changing Base to Common Logs
Use common logarithms to evaluate log5 140.
Example Changing Base to Natural Logs
Use natural logarithms to evaluate log5 140.
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Example

• Use logarithms to evaluate log37.
• Solution

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