λόγος + ἀριθμός

Logarithm was coined in 1614 by the Scottish mathematician John Napier of Murchiston in his treatise Mirifici logarithmorum canonis descriptio — 'A Description of the Wonderful Canon of Logarithms.' Napier constructed the word from Greek λόγος (lógos, 'ratio, reason, word') and ἀριθμός (arithmós, 'number'), producing 'ratio-number' or 'reasoning-number.' The lógos in question referred to the mathematical relationship Napier had discovered: that a sequence of numbers in geometric progression corresponds, term by term, to a sequence of numbers in arithmetic progression. The logarithm of a number was the index in the arithmetic sequence that corresponded to that number in the geometric sequence — a ratio captured in a number. Napier was describing a relationship between two ways of counting, and he chose Greek roots to signal that this was a fundamental mathematical concept, not a mere computational trick.

Napier's motivation was practical and urgently needed by the astronomers and navigators of his era. Multiplying large multi-digit numbers — the calculations required to predict planetary positions, navigate by the stars, or compute artillery trajectories — was enormously time-consuming and error-prone. Napier's key insight was that multiplication could be converted into addition: log(a × b) = log(a) + log(b). If you had tables of logarithms, you could multiply two large numbers by looking up their logarithms, adding the logarithms, and looking up the antilogarithm of the sum. A calculation that previously required hours of long multiplication could be done in minutes. Napier spent twenty years computing his logarithm tables by hand before publishing. His tables were immediately adopted by astronomers across Europe; the astronomer Johannes Kepler, who was struggling with exactly such computations, wrote that the tables saved him months of work.

The mathematician Henry Briggs, who met Napier in Edinburgh and worked with him in the last years of Napier's life, proposed and calculated a modification: common logarithms based on powers of ten (log₁₀), rather than Napier's original base. Briggs published tables of common logarithms in 1617 (Napier died in 1617 before seeing them complete) and extended them to fourteen decimal places in 1624. The Briggsian or common logarithm became the standard computational tool for three centuries of science and engineering. Slide rules — the mechanical analog computers that engineers and scientists carried until the 1970s — operated on logarithmic scales, converting multiplication to the physical addition of lengths. Every calculation that put satellites in orbit and men on the moon was performed, in the decades before electronic computers, with logarithm tables and slide rules.

The electronic calculator's arrival in the 1970s made logarithm tables obsolete almost overnight for practical computation. But logarithms remained essential as mathematical objects — functions of fundamental importance to analysis, information theory, and physics. The natural logarithm (base e, the mathematical constant approximately 2.71828) turned out to be deeply embedded in the mathematics of growth, decay, and probability. The Richter scale, the decibel scale, the pH scale, the musical scale of octaves, the formula for entropy, the distribution of prime numbers, the time-complexity of efficient algorithms — all involve logarithms. The ratio-number that Napier invented to save astronomers from arithmetic has turned out to be one of the fundamental functions by which the universe organizes itself.