Solving Logarithmic Equations Math Example 2

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Example 2

medium
Solve log⁡(x+3)+log⁡(x−1)=log⁡(5)\log(x+3) + \log(x-1) = \log(5).

Solution

  1. 1
    Use the product rule: log⁡[(x+3)(x−1)]=log⁡(5)\log[(x+3)(x-1)] = \log(5).
  2. 2
    Since the logarithms are equal: (x+3)(x−1)=5(x+3)(x-1) = 5.
  3. 3
    Expand: x2+2x−3=5x^2 + 2x - 3 = 5, so x2+2x−8=0x^2 + 2x - 8 = 0.
  4. 4
    Factor: (x+4)(x−2)=0(x+4)(x-2) = 0, giving x=−4x = -4 or x=2x = 2.
  5. 5
    Check domains: x=−4x = -4 makes log⁡(x−1)=log⁡(−5)\log(x-1) = \log(-5) undefined. So x=2x = 2 is the only solution.

Answer

x=2x = 2
When solving logarithmic equations, always combine logs using properties, convert to an algebraic equation, solve, and then check that all solutions keep every logarithm argument positive. Extraneous solutions are common in logarithmic equations.

About Solving Logarithmic Equations

Solving equations containing logarithms by converting to exponential form or using log properties to combine and simplify.

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More Solving Logarithmic Equations Examples

Example 1 easy

Solve [formula].

Example 3 medium

Solve [formula].

Example 4 hard

Solve [formula].

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