Number · logarithms

Logarithms and the slide rule

The logarithm of a number is the power of 10 that gives it: log 1000 = 3, log 2 ≈ 0.301. Because powers add when numbers multiply, a table of logarithms turns every multiplication into an addition, and every root into a division. A slide rule does the same addition with two rulers laid side by side.

Take a logarithm

Characteristic and mantissa

A logarithm splits into two parts. The whole-number characteristic says where the decimal point is; the decimal mantissa depends only on the digits. So 3.456, 345.6 and 0.003 456 share the mantissa 0.5386, and a printed table needed only the mantissas of numbers from 1 to 10.

For numbers below 1 the tables kept the mantissa positive and wrote a negative characteristic with a bar over it: log 0.0345 = −1.4622 was written 2̄.5378, meaning −2 + 0.5378.

A working slide rule

Drag the middle strip (the slide) and the hairline (the cursor), or use the examples. Distances on the C and D scales are proportional to logarithms, so sliding C along D adds logarithms: to multiply 2 × 3, put the 1 of C over 2 on D and read D under 3 on C.

Scale	Range	Position along the rule	Used for
D, C	1 – 10	log x	multiplication and division
CI	10 – 1	1 − log x	reciprocals
A, B	1 – 100	½ log x	squares and square roots
K	1 – 1000	⅓ log x	cubes and cube roots
L	0 – 1	x	the mantissa itself: log of the D reading

A slide rule gives three significant figures; you place the decimal point yourself, which is exactly what the characteristic does.

Then and now

John Napier published logarithms in 1614 and Henry Briggs, with Napier’s agreement, recast them in base 10 within a few years. William Oughtred described logarithmic scales that move against each other in The Circles of Proportion (1632). For the next 340 years slide rules and log tables did the arithmetic of ships, bridges, aircraft and the first spacecraft, until pocket electronic calculators replaced them in the 1970s.