Logarithm Properties

Pre-Calculus · Axiom Academy

Three rules turn multiplication into addition — because a logarithm is just an exponent in disguise. The log of a product is the sum of the logs. The reason is the exponent law : a logarithm is an exponent, so multiplying the numbers means adding their exponents. multiply inside → add the logs The log of a quotient is the difference of the logs. Dividing subtracts exponents — b^x / b^y = b^ x-y — so the log of a ratio is the log of the top minus the log of the bottom. divide inside → subtract the logs Dividing subtracts exponents: 5 - 2 = 3 . Since 2^3 = 8 , that leftover 3 is precisely . Same machinery as the product rule, run in reverse. The log of a number raised to a power brings the exponent out front as a multiplier . A power is repeated multiplication, and the product rule turns each repeat into a repeated addition of the same log — which is just multiplication. exponent inside → coefficient out front , the exponent that makes 2^3 = 8 . 4. Two Special Values & One Trap Two values fall straight out of the definition, and one tempting "rule" is flat-out false. Knowing all three keeps you from the most common logarithm mistakes. You've seen all three logarithm properties as the exponent laws they come from — and the one trap to avoid. Scroll up to revisit any step.

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