Solving Logarithmic Equations

Pre-Calculus · Axiom Academy

LESSON Solving Logarithmic Equations Turn a logarithm into an exponential, solve for x — then check every answer, because the argument of a log must stay positive. 1. A Log Is an Exponent in Disguise Every logarithm statement is an exponential statement wearing different clothes. The base stays the base; the value the log returns is the exponent ; and what's inside the log is the result of raising the base to that exponent. 2. One Log Alone: Convert to Exponential When a single logarithm is already isolated, you're one step from the answer. Drop the value on the right down to become the exponent on the base, and the equation solves itself. 1 Rewrite in exponential form — the 4 becomes the exponent on base 2 . 2 Evaluate. The argument is , so the answer is valid. 3. Several Logs on One Side: Condense First If two logs of the same base are added, the product rule fuses them into one: . Once there's a single log again, you're back to the move from Step 2 — convert to exponential. 1 Condense with the product rule. 2 Convert to exponential form. 3 Solve the quadratic, then keep only roots that make every argument positive. 4. Logs on Both Sides: Drop the Logs When a single log of the same base sits on each side, you don't need an exponential at all. Because is one-to-one, equal logs force equal arguments — the matching log "caps" lift off together. 1 Equal logs ⇒ equal arguments. 2 Solve, then check: at x=4 both arguments equal . 5. The Step Everyone Skips: Check the Domain

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