The arithmetic hidden inside multiplication
Multiplication can be slow because each digit creates work: multiplying two multi-digit numbers requires partial products, carrying, and careful addition. Logarithms offered a way around much of that effort by giving numbers a different kind of address.
In modern terms, a logarithm tells you what power of a chosen base produces a number. For example, in base 10, the logarithm of 100 is 2 because 10 × 10 = 100. The crucial rule is that multiplying numbers adds their logarithms: the logarithm of a product equals the sum of the logarithms of its factors.
A table did the conversion
A calculator-free user first found each factor in a printed table and noted its logarithm. After adding the two entries, the user searched the same table for that sum; the number beside it was the product. The table effectively translated numbers into a form that was easier to combine, then translated the result back.
The method shifted much of the work from the person doing a calculation to the people who prepared and checked the tables. Tables could contain rounding errors, and values between listed entries might require interpolation—estimating a result between known entries—but for many calculations this was still quicker than multiplying by hand.
Napier’s route to the shortcut
John Napier published his logarithm system in 1614. His original numbers were constructed differently from the familiar base-10 logarithms taught today, but they served the same practical aim: pairing a progression of numbers with a simpler progression that made difficult calculations easier. Later refinements made logarithm tables more convenient for everyday arithmetic.
For generations, the payoff was tangible: a navigator or astronomer could replace a long multiplication with a table lookup and an addition. The hard operation had not vanished; its labor had been compressed into a carefully prepared page of numbers.
