AI Tutor
Imagine finishing a page-long algebra problem, boxing your answer of x=−2x = -2, and feeling confident — only for a calculator to flash an immediate DOMAIN ERROR.
In a logarithmic expression like log⁡b(A)\log_b(A), the input AA is called the argument. Because a positive base bb raised to any real power is always positive, the argument AA must be strictly greater than zero.
Algebraic steps like squaring or combining logs can create extraneous solutions — values produced by correct algebra that fail in the starting equation. How do we spot them every single time?

The Two-Step Substitution Test

To verify any candidate solution, you must substitute it directly into the original equation — before any properties were applied. You then verify two separate conditions: every argument must be positive, and the left-hand side must equal the right-hand side.
Let's see this test in action on a concrete equation with two algebraic candidates.

Verifying a Valid Solution

Suppose solving log⁡2(x)+log⁡2(x−2)=3\log_2(x) + \log_2(x - 2) = 3 yielded two candidates: x=4x = 4 and x=−2x = -2. Let's test x=4x = 4 first.
First, check the arguments: in log⁡2(4)\log_2(4), 4>04 > 0; in log⁡2(4−2)=log⁡2(2)\log_2(4 - 2) = \log_2(2), 2>02 > 0. Both arguments are strictly positive, so we proceed to calculate.
Because 3=33 = 3, x=4x = 4 is a verified solution. But what happens when we test our second candidate, x=−2x = -2?