Logarithm of a Quotient

The mirror image of the product rule: since $b^m \div b^n = b^{m-n}$, dividing inside a logarithm subtracts outside it. Worked example: log₃(81/3) = log₃81 − log₃3 = 4 − 1 = 3, which is log₃27, as it should be. Solve it the other way and you get the everyday version — if log₁₀(x/4) = 2 then x/4 = 100, so x = 400.

Subtracted logs are how science reports ratios. A pH is −log₁₀ of the hydrogen-ion concentration, so a solution one pH unit lower is exactly ten times more acidic; the decibel scale compares a sound's intensity with a reference intensity by the same subtraction, and stellar magnitudes do it for brightness. Two traps: log(x − y) is not log x − log y, and log x / log y is not log(x/y) — that quotient of logs is the change-of-base formula, an entirely different animal. It follows too that log(1/y) = −log y, which is why very small concentrations produce large positive pH values.