Solving Exponential and Logarithmic Equations

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Section 4.5

• Solving Exponential and Logarithmic Equations

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Solving Exponential Equations

• Equations with variables in the exponents, such
  as 3x 40 and 53x 25, are called exponential
  equations.

Base-Exponent Property For any a gt 0, a ? 1,
ax ay ? x y.
3
Example

• Solve

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• Check 52x ? 3 125
• 52(3) ? 3 ? 125
• 53 ? 125
• 125 125 True
• The solution is 3.

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Another Property

• Property of Logarithmic Equality
• For any M gt 0, N gt 0, a gt 0, and a ? 1,
• loga M loga N ? M N.

6
Example

• Solve 2x 50

7
Example

• Solve e?0.25w 12

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Solving Logarithmic Equations

• Equations containing variables in logarithmic
  expressions, such as log2 x 16 and
  log x log (x 4) 1, are
  called logarithmic equations.
• To solve logarithmic equations algebraically, we
  first try to obtain a single logarithmic
  expression on one side and then write an
  equivalent exponential equation.

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Example

• Solve log4 x ?3

10

• Check log4 x ?3
• The solution is

11
Example

• Solve

12
Check Solution to Example
Check For x 3 For x ?3
Negative numbers do not have real-number
logarithms. The solution is 3.
13
Example

• Solve

The value 6 checks and is the solution.