Why Logarithms Matter — Turning Multiplication into Addition

Fourteen.

Each fold doubles it, so fourteen gives 16,384 layers — about 1.6 metres. Twenty-three folds reach 838 metres, taller than any building. Forty-two folds reach the Moon. One sheet of paper, forty-two folds.

So logarithms turn up wherever something grows or shrinks by repeated multiplication: interest on savings, a spreading infection, a radioactive substance halving, coffee cooling.

But why does that make arithmetic easier?

Two multiplied by itself four times is 16; six times is 64. So what is 16 times 64?

You can answer without multiplying. Four doublings times six doublings is ten doublings. Four plus six is ten, and ten doublings gives 1,024.

The catch was that the trick only worked on tidy numbers — 2, 4, 8, 16. Three, seven and 1234 are not whole numbers of doublings. Filling in everything between took one man twenty years.

A little further in

Think about a sixteenth-century astronomer's day. To compute a planet's position you multiply an eight-digit number by another eight-digit number. By hand.

That is sixty-four single-digit multiplications, written in aligned rows and summed. Fifteen minutes or so, and one wrong digit anywhere means starting again. Such calculations were needed dozens of times a day.

The idea

Napier was a Scottish laird for whom mathematics was not the day job. He gave twenty years to this problem.

The core idea: multiply two by itself repeatedly and you get 2, 4, 8, 16, 32. Now multiply 4 by 8 and you get 32. But 4 is two doublings, 8 is three, and 32 is five. Two plus three is five. Multiplication has become addition of exponents.

The difficulty was that this works only for 2, 4, 8 and their kin. Three, seven, 1234 are not whole numbers of doublings. Napier filled in the gaps finely enough to tabulate, for every number, how many doublings it amounts to.

What actually happened

The tables appeared in 1614, and the response was immediate.

Kepler had spent years grinding through calculations to pin down the orbit of Mars. He took up the tables at once, and the planetary tables he published in 1627 were computed with logarithms. They were more accurate than anything before them.

It doubled the life of the astronomer.attributed to Laplace, on logarithms

Not an exaggeration. With that much time removed from computation, twice the work fits in one lifetime. For the next three centuries, until electronic calculators, every precise calculation ran on log tables or a slide rule. Apollo engineers still carried one.

Beyond a tool

Euler turned the thing around. Napier had built tables for converting multiplication into addition; Euler made clear that the logarithm is the inverse of the exponential. A calculating trick became a function.

With that, logarithms left the toolbox and became a language for describing nature — because whenever a quantity grows or shrinks in proportion to its own size (interest, populations, radioactive decay, cooling coffee), the logarithm is the natural way to write the change.

Human senses work much the same way. Add a candle to a room lit by one and the change is obvious; add one to a room lit by a hundred and nobody notices. What we perceive is not difference but ratio.

The question that remainsTurning multiplication into addition made the work easy. What that we find hard today would become easy if we changed the scale?