Solving Logarithmic Equations

Strategy 1: Use One-to-One Property

If $\log_b(x) = \log_b(y)$, then $x = y$

Example: $\log(x + 3) = \log(2x - 1)$ $x + 3 = 2x - 1$ $x = 4$

Strategy 2: Convert to Exponential

Use the definition: $\log_b(x) = y$ means $b^y = x$

Example: $\log_2(x) = 5$ $x = 2^5 = 32$

Strategy 3: Condense First

Use log properties to combine:

• Product: $\log(a) + \log(b) = \log(ab)$
• Quotient: $\log(a) - \log(b) = \log(\frac{a}{b})$
• Power: $n\log(a) = \log(a^n)$

Check Your Answers!

Logarithms require positive arguments.

Always verify solutions don't make any log argument ≤ 0.

Common Equation Types

Type 1: $\log_b(x) = c$ → $x = b^c$

Type 2: $\log_b(x) = \log_b(y)$ → $x = y$

Type 3: $\log_b(x) + \log_b(y) = c$ → $\log_b(xy) = c$ → $xy = b^c$

Example

Solve: $\log_3(x + 1) + \log_3(x - 1) = 2$

Step 1: Use product property $\log_3[(x + 1)(x - 1)] = 2$

Step 2: Convert to exponential $(x + 1)(x - 1) = 3^2 = 9$

Step 3: Solve $x^2 - 1 = 9$ $x^2 = 10$ $x = \pm\sqrt{10}$

Step 4: Check domain Only $x = \sqrt{10}$ makes both logs positive!