Logarithm Properties & Equations

Calculus Readiness · Axiom Academy

The rules that turn multiplication into addition, division into subtraction, and exponents into multipliers — watch each one happen. 1. The Product Rule — Multiplication Becomes Addition The single most important log law: the log of a product is the sum of the logs. The two factors that are multiplied inside the log break apart into two logs that are added outside it. Watch the 8 · 4 inside split into log 8 + log 4 . because powers add when you multiply: 2. The Quotient Rule — Division Becomes Subtraction The mirror image of the product rule. A quotient inside the log breaks into a difference of logs — the numerator's log minus the denominator's log. The order matters here: top minus bottom, never the other way around. because powers subtract when you divide: 3. The Power Rule — The Exponent Drops to the Front An exponent trapped inside a log doesn't have to stay there. The power rule lets it slide down and become a multiplier out front . Watch the exponent on 8^2 detach from the argument and drop down to the left of the log. because , and the product rule turns those into n equal logs A root is just a fractional power, so the same rule handles it: For example, Any convenient base works — base 10 and base e are just the buttons you have. 4. Expanding & Condensing — Running the Rules Both Ways

This is the written version of the interactive lesson above. See the full Calculus Readiness course.