Log Equations and Extraneous Solutions

A logarithm only exists for a positive argument, so log(x) demands x > 0. When you solve a log equation, you usually combine the logs with log rules, then exponentiate both sides to strip the log away. That exponentiating step is legal algebra, but it has a side effect: it creates an equation, often quadratic, whose solution set can be bigger than the original. The new equation does not know about the domain restrictions the original logs carried with them. So it can hand you a root that solves the simplified equation perfectly while being completely illegal in the equation you started with.

Raising the base to both sides works because exponentials and logs are inverses: if log_b(A) = c, then b^c = A, because b^log_b(A) = A. That is the move that clears the log and leaves ordinary algebra, usually a linear or quadratic equation to solve by factoring or the quadratic formula.

Take log(x) + log(x - 3) = 1. Combine: log(x(x-3)) = 1, so x(x-3) = 10¹ = 10, giving x² - 3x - 10 = 0, which factors as (x-5)(x+2) = 0, so x = 5 or x = -2. Check both in the original equation: x = -2 makes log(x) = log(-2), the log of a negative number, undefined. Only x = 5 survives. This is exactly the gate every log-equation answer must pass through before you trust it, and it is the same domain thinking that governs where exponential and log graphs live, feeding directly into solving exponential equations and modeling with both function families.