Can multiplication be turned into addition?

That time, that place

Sixteenth-century navigators and astronomers multiplied and divided ten-digit numbers by hand. One multiplication could take half a day, and finding the slip took another.

A slip meant a ship arriving in the wrong place.

Why this question

Napier seized on this. Take 2, 4, 8, 16, 32 — each double the last — and number them 1, 2, 3, 4, 5.

Then 8 times 4 becomes: entry 3 and entry 2, add to get 5, and entry 5 is 32. Multiplication has turned into addition.

The remaining task was to make the table fine enough. He spent twenty years on it.

What was found

The tables appeared in 1614. Kepler wrote that they cut his computing time by more than half. Laplace said logarithms doubled an astronomer's life.

The slide rule followed and stayed an engineer's tool for three centuries — one flew on Apollo.

And logarithms are still here. Decibels, earthquake magnitude and pH are all log scales: a way of folding a range too wide to handle into one a person can read.