Logarithms Explained: How the Slide Rule Multiplies by Adding
A logarithm is the power a fixed base must be raised to in order to give a number, and that rule let the slide rule turn multiplication into addition.
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A logarithm answers one question: to what power must a fixed base be raised to produce a given number? In base 10, the logarithm of 1000 is 3, because 10³ = 1000. Logarithms turn multiplication into addition, and that property made them the basis of hand calculation for more than three centuries, until electronic calculators replaced the slide rule in the 1970s.
What is a logarithm?
Start with powers of 10. Since 10² = 100, the base-10 logarithm of 100 is 2. Since 10³ = 1000, the logarithm of 1000 is 3. The written form is log₁₀ 100 = 2, which reads "the logarithm of 100 to base 10 is 2". The same pattern works for any positive base, and base 2 and base e are common in mathematics.
The useful rule comes from the way powers combine. Multiplying two powers of the same base adds their exponents: 10² × 10³ = 10⁵. A logarithm carries that rule to any pair of numbers, so log (a × b) = log a + log b. A multiplication of two numbers becomes an addition of their logarithms, which is far easier to do by hand from a table.
How did logarithm tables work?
Before calculators, people used printed tables. The Scottish mathematician John Napier published logarithms in 1614, and his original tables were not based on 10. The English mathematician Henry Briggs produced the base-10 tables that became standard. A table listed the logarithm of many numbers, so a user could look up two logarithms, add them, and then look up the number whose logarithm was the sum.
This was a large saving of effort. A long multiplication with many digits became two lookups and one addition. The method was only as good as the printed table, and errors in published tables were a known problem. Compiling a table by hand was slow, and a single wrong entry could spread through later copies.
How did the slide rule use logarithms?
A slide rule has scales marked with logarithms rather than with evenly spaced numbers. On a log scale, the distance from 1 to a number is proportional to its logarithm, so the distance from 1 to 2 is a fixed length and the distance from 1 to 3 is longer by a different fixed amount. Two such scales slide past each other. Moving one along the other adds lengths, and adding lengths adds logarithms, which multiplies the numbers.
To multiply 2 by 3, the user lines up the 1 on the sliding scale with the 2 on the fixed scale. The 3 on the sliding scale then sits above 6 on the fixed scale, which is the product. Division works in reverse. The slide rule does the adding by physical length, so its answer is only as exact as the eye can read the marks. Most slide rules were accurate to about three significant figures, which was enough for much engineering work.
The scales were developed in the 1620s. Edmund Gunter, an English mathematician, put a logarithmic scale on a single line, and William Oughtred combined two such scales into a sliding pair, which is the basic form of the slide rule.
Why did the slide rule disappear?
Electronic calculators made the slide rule obsolete quickly. The Hewlett-Packard HP-35, released in 1972, is often cited as the first handheld scientific calculator, and it could compute logarithms directly. Within a few years, engineers and students who had used slide rules for decades had switched to pocket calculators.
The logarithm did not disappear with the slide rule. It remains a basic tool in science, for measuring earthquakes, sound levels and the acidity of solutions, and it appears in the analysis of algorithms. A search that halves the list at each step needs about log₂ of the list length steps, which is why binary search is fast even for very long lists. The big O notation that describes such growth is built on the same idea.
Earlier aids such as the abacus sped up the handling of digits. The logarithm worked differently, because it turned a hard operation into an easy one.
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arithmetic calculation history of mathematics logarithms