quotient law of logarithms

Mirroring how the product law turns multiplication into addition, the quotient law turns division into subtraction. To find the log of one number divided by another, subtract the logs. It is the natural partner of the product law, and together they let logs replace hard arithmetic with easy arithmetic.

The formula is log_b(M / N) = log_b(M) - log_b(N), for any allowed base b and positive M and N. Once again the reason is an exponent rule: dividing powers of the same base subtracts their exponents, and logs are exponents. So dividing the numbers shows up as subtracting their logs.

A handy special case: since log_b(1) = 0, the law gives log_b(1 / N) = -log_b(N). That is the log version of saying a reciprocal flips the sign of the exponent. As with the product law, the rule applies to a quotient inside one log, not to log(M) / log(N), which is a different quantity entirely.