Solving Logarithmic Equations

Solving a logarithmic equation means condensing to a single log, converting to exponential form, solving, and auditing every candidate against the logs' domains in the original.

Solve log₂(x) + log₂(x − 2) = 3. Condense by the product rule: log₂(x² − 2x) = 3. Convert costumes: x² − 2x = 2³ = 8. Solve the quadratic: x² − 2x − 8 = 0 factors to (x − 4)(x + 2), candidates 4 and −2. Now the audit, in the original: at x = 4, both logs read positive inputs — log₂(4) + log₂(2) = 2 + 1 = 3, true. At x = −2, the very first log asks for log₂(−2), which no exponent of 2 can answer. Discard. One solution: x = 4. The extraneous candidate was manufactured by the condensing, exactly as squaring manufactured them before.