Logarithm Change of Base

Calculators carry two logarithm buttons, base 10 and base e, but questions arrive in every base — so change of base rewrites $\log_b x$ as a ratio of logs you can actually press. Any common base works, because the conversion factors cancel: log₂32 = ln 32 / ln 2 = log₁₀32 / log₁₀2 = 5 either way. Solving in the other direction identifies an unknown base: if $\log_b 1000 = 3$, then $b = 1000^{1/3} = 10$.

Logarithms exist for exactly this reason — to trade hard arithmetic for easy arithmetic. John Napier published Mirifici Logarithmorum Canonis Descriptio in 1614 after twenty years of hand calculation, and Laplace later said the invention, "by shortening the labours, doubled the life of the astronomer." Every slide rule, every log table, and every decibel, pH and Richter reading descends from it. The traps are domain traps: x must be strictly positive (no real power of a positive base ever gives zero or a negative), and the base must be positive and not 1. Note also which quantity goes where: $\log_b x$ is ln x over ln b, argument on top; flipping them gives the reciprocal, so log₂8 = 3 becomes 0.333 if you invert.