Napier’s logarithms turned multiplication into addition
A logarithm asks what exponent the base needs to make a number—so log₁₀(1000) = 3. John Napier introduced them in 1614 to ease calculation; tables and slide rules followed. Euler later tied logs to the exponential and popularized base e ≈ 2.718 for natural logs.
Napier published his method in 1614 in a book whose title translates as Description of the Wonderful Canon of Logarithms. He was not alone: Jost Bürgi had developed tables of progressions around 1600, and the trigonometric trick called prosthaphaeresis also existed. Navigators, surveyors, engineers and scientists took up the new tables quickly, because a laborious multi-digit multiplication became two look-ups and an addition, and division became subtraction. Before computers, that saving mattered enormously.
Three bases dominate. Base 10, the common logarithm, suits hand work in the decimal system: it tells you how many digits a whole number has, since the digit count is the first integer above its log, so 5986, with a log near 3.78, has four digits. Base 2 rules computer science, and music theory measures intervals in cents, 1200 times the binary log of a pitch ratio, while photographers count exposure in stops on a rescaled binary scale. Information theory uses the natural and binary logs for nats and bits. In pure mathematics an unadorned log usually means base e, in computing it often means base 2, and in engineering it frequently still means base 10.
A few identities do most of the work. The log of any base taken to that same base is 1, and the log of 1 is 0. Logs of ratios become differences, the log of a p-th power is p times the log, and a p-th root divides the log by p. Changing base just means dividing by the log of the new base, which is how calculators with only base 10 and base e keys handle any base. For instance, the base-2 log of 16 is 4, of one half is minus 1, and the common log of 150 is about 2.176.
Beyond positive reals the idea becomes multi-valued. The complex logarithm inverts the complex exponential with many branches, and the discrete logarithm in finite groups underpins public-key cryptography. Everyday scales such as decibels and pH are logarithmic too.
Source: Logarithm
