Advanced Algebra

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Advanced Algebra

• Lesson 5-9
• Properties of Logarithms

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• There are three properties of logarithms that
  come directly from the properties of exponents.
• The log of a composite number (a number having
  factors) such as 15 can be found by adding the
  logs of the two factors, 5 and 3.
• log 5 0.698970.
• log 3 0.477120.
• log 15 1.176981..

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• In other words,
  . The property is simply an example
  of the fact that logarithms are exponents. When
  you multiply two powers with equal bases, you add
  their exponents.
• A similar property applies to the log of a
  quotient.
• log 30 1.477121.
• log 5 0.698970..
• log 6 0.778151.

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• In other words,
  
• Again from exponents, when you divide powers
  with equal bases, you subtract their exponents.
• The third property applies to the log of a
  power. We know that 32 is equal to . We can
  get log 32 by multiplying log 2 by 5.
• log 32 1.505149.
• 5(log 2) 5(.301029) 1.505149

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• In other words
  . This can be explained by when you raise a
  power to a power, you multiply the exponents.
• Formal Properties of Logarithms
• Logarithm of a Product
• The log of a product equals the sum of the
  logs of the two factors.

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• Logarithm of a Quotient
• The log of a quotient equals the log of the
  numerator minus the log of the denominator.
• Logarithm of a Power
• The log of a power equals the exponent
  times the log of the base.

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Examples

• 1) Given and
  .
• Find log 48
• 2) Express as a
  single logarithm.

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• 3) Solve by taking the log of
  each member and using the log of a property.