Logarithms

A logarithm answers the question "what power must I raise the base to, to get this number?" Because exponents add when powers are multiplied — 10² × 10³ = 10⁵, and 2 + 3 = 5 — logarithms turn every multiplication into an addition, every division into a subtraction, every root into a division. John Napier of Merchiston published the first tables in 1614 in *Mirifici Logarithmorum Canonis Descriptio*, "A Description of the Wonderful Canon of Logarithms," 90 pages of tables preceded by 57 pages explaining how to use them. He had worked on them for two decades.

In 1614 the frontier science was astronomy, and astronomy meant multiplying seven-digit trigonometric values by hand, over and over — slow, and a fresh chance to slip at every digit. Logarithms replaced the worst operation, multiplication, with the easiest, addition. Kepler, then wrestling the orbit of Mars out of raw observations, seized on Napier's tables and dedicated a work to him; the computations that had cost him years were suddenly tractable. For three and a half centuries, until the electronic calculator, logarithm tables and their mechanical cousin the slide rule were the way serious numerical work got done.

Napier's own construction was geometric and slightly awkward — he imagined two points moving along lines, one at constant speed and one slowing in proportion to the distance remaining, and defined the logarithm from their relation. His base was close to 1/e and his logarithms ran the "wrong" way for everyday arithmetic. The Oxford geometer Henry Briggs saw the tables, grasped their power, and travelled to Edinburgh in 1616 to propose a repair: anchor the system so that the logarithm of 1 is 0 and of 10 is 1. These base-10 "common logarithms," which Briggs then computed to fourteen places, are the ones that filled the tables and slide rules of the next three centuries.

Logarithms were the calculating engine of science and engineering from the 1600s to the 1970s — every navigator, surveyor, physicist, and engineer worked through log tables or a slide rule. The idea reaches far past computation: the logarithmic scale is how we measure earthquakes (Richter), sound (decibels), and acidity (pH), because human senses and many natural spans are multiplicative. The constant *e* and the natural logarithm that emerged from this work sit at the centre of calculus, compound growth, and probability — Napier's labour-saving trick turned out to be a fundamental shape of nature.