Solving Logarithmic Equations

Definition of a Logarithm

• Logarithmic form: y = logₐ(x)
• Equivalent Exponential form: x = aʸ
• Conditions:
  • a > 0
  • a ≠ 1
• Equality Property: If logₐ(m) = logₐ(n), then m = n
  • m, n, and a must be positive
  • a cannot be 1
  • When solving, check the solutions to ensure they are positive
  • Negative results may be extraneous solutions

Solving Logarithmic Equations

1. Single Logarithmic Expression Equal to a Constant

• Example: log₂(5x) = 4
  • Convert to exponential form: 2⁴ = 5x
  • Solve: x = 16/5
  • Check: Insert back into the equation to confirm positivity
  • Result: x = 16/5 is valid since log₂(16) = 4

2. Logarithmic Expression Equal to Another Logarithmic Expression

• Example: 2 log₅(x) = 3 log₅(4)
  • Convert coefficients to exponents using logarithmic properties
  • log₅(x²) = log₅(4³)
  • Set expressions equal: x² = 64
  • Solve: x = ±8
  • Check:
    • x = 8 is valid
    • x = -8 is extraneous (log of negative is undefined)
  • Result: x = 8 (only solution), x = -8 (extraneous)

3. Logarithm of Sum Equal to a Constant

• Example: log(x) + log(x + 15) = 2
  • Use log property: log(x(x + 15)) = 2
  • Solve: x(x + 15) = 10²
  • Quadratic form: x² + 15x - 100 = 0
  • Factor: (x + 20)(x - 5) = 0
    • Solutions: x = -20, x = 5
  • Check:
    • x = 5 is valid
    • x = -20 is extraneous (log of negative is undefined)
  • Result: x = 5 (only solution), x = -20 (extraneous)

4. Logarithm of Sum with Different Bases

• Example: log₂((x + 7)(x + 8)) = 1
  • Use log property: log₂(x² + 15x + 56) = 1
  • Convert to exponential form: x² + 15x + 56 = 2
  • Solve: x² + 15x + 54 = 0
  • Factor: (x + 9)(x + 6) = 0
    • Solutions: x = -9, x = -6
  • Check:
    • x = -6 is valid (results in positive values in original log expressions)
    • x = -9 is extraneous (results in negative values)
  • Result: x = -6 (only solution), x = -9 (extraneous)