product law of logarithms

Logarithms have a magic trick: they turn multiplication into addition. If you want the log of two numbers multiplied together, you can just add their separate logs. This is the product law of logarithms, and it is the reason logarithms once made long multiplications easy by hand.

Written as a formula, log_b(M · N) = log_b(M) + log_b(N), valid for any allowed base b and any positive M and N. The reason is exactly the exponent rule underneath: when you multiply powers with the same base you add the exponents, and a logarithm is just an exponent. So the addition of logs mirrors the multiplication of the numbers.

Use it carefully in both directions: you may split log_b(MN) into a sum, or combine a sum of logs back into a single log of a product. But note the law applies to a product inside one log — it says nothing about log(M) · log(N), and there is no simple rule for the log of a sum, log_b(M + N).