Logarithm of a Product

This is the identity logarithms were invented for. Because $b^m \times b^n = b^{m+n}$, taking logs of a product adds the logs of its factors: log₁₀(20 × 50) = log₁₀20 + log₁₀50 = 1.301 + 1.699 = 3, which is log₁₀1000. Multiplying two awkward numbers becomes looking up two logs, adding, and looking the answer back up.

Sixteenth-century astronomers were doing exactly that trick already, using a trigonometric identity called prosthaphaeresis to convert products into sums of cosines; Napier's tables in 1614 made it general, and Henry Briggs's base-10 recasting a few years later put the method in every navigator's chart room for the next three and a half centuries. The traps are the ones every algebra course flags: log(x + y) is not log x + log y, and log x × log y is not log(xy) either. The rule converts multiplication inside the log into addition outside it, never the reverse. Both factors must also be positive; log(−4) + log(−9) is undefined even though log 36 is perfectly fine.