Logarithms

The logarithm as a calculator

Before the logarithm became an abstract function in the school syllabus, it was above all a tool – a fast way to calculate. Before calculators existed, astronomers, navigators and mathematicians had to multiply and divide many-digit numbers by hand, and that was slow and error-prone. The logarithm gets around the problem with a single trick: it turns multiplication into addition.

Try it yourself. Add up these numbers in a column:

The addition takes under two minutes. Now try multiplying the same seven numbers together – even with a calculator at hand it will take much longer, and by hand it would be hours of drudgery.

This difference is exactly why the logarithm was invented. If there were a table in which you could find the “logarithm” of every number, you would only need to add the logarithms and then look up in the table which number the sum corresponds to – the result of the multiplication, without multiplying even once.

John Napier

John Napier (1550–1617) was a Scottish mathematician, physicist, astronomer and landowner from Merchiston near Edinburgh. Logarithms were a side interest for him – he made his living mainly by running his estate and writing theological works. In mathematics, though, he was immensely persistent: according to the surviving sources he worked on his system of logarithms for about twenty years before describing it in 1614 in the book Mirifici Logarithmorum Canonis Descriptio (A Description of the Wonderful Rule of Logarithms). The book was received with immediate enthusiasm by the mathematicians of the day.

Besides logarithms, Napier is known for inventing Napier’s bones – a mechanical aid for multiplication – and for helping to spread the use of the decimal point in writing numbers.

Arithmetic and geometric sequences

To “make” the logarithm, Napier needed to connect two worlds: the world of multiplication, where numbers grow by being multiplied by the same number again and again, and the world of addition, where numbers grow by having the same number added again and again.

Geometric sequence – multiply by the same number each time: 1, 2, 4, 8, 16, 32, 64…

Each new term is obtained by multiplying the previous one by the same number, the common ratio q.

1, 3, 9, 27, 81… (q = 3)
64, 32, 16, 8, 4… (q = ½)

Arithmetic sequence – add the same number each time: 0, 1, 2, 3, 4, 5, 6…

Each new term is obtained by adding the same number, the common difference d, to the previous one.

3, 7, 11, 15, 19… (d = 4)
20, 17, 14, 11, 8… (d = −3)

When you write these two rows side by side, line by line, something useful happens:

Want to multiply 8 × 32? Find 8 and 32 in the left column, read off their “partners” on the right (3 and 5), add them (3 + 5 = 8) and find the number in the left column whose partner on the right is the result, 8: it is 256. And indeed, 8 × 32 = 256.

The same table can also divide – instead of adding the right-hand numbers, you subtract them – and take roots, where you divide the number on the right by the degree of the root.

This table is in fact a prototype of a table of logarithms: the left column holds the “numbers”, the right column their logarithms to base 2.

Where the word “logarithm” came from

At first Napier did not call his new numbers logarithms. In the manuscript we now know as the Constructio, he called them artificial numbers, because they are numbers that the mathematician “makes” himself to lighten the work, not numbers that directly measure anything. Only later, in the Descriptio, did he come up with a new name: logarithm, made of two Greek words – logos (ratio, word, reckoning) and arithmos (number). So a logarithm is literally a “ratio number”: a number that expresses a ratio, that is, how many times the base has to be multiplied.

The problem of choosing a base

Our table of powers of two (1, 2, 4, 8, 16…) is nice as a demonstration, but useless for practical calculation. Almost all numbers are missing from it – there is no row for 3, 5, 6, 7… And the further down the table you go, the bigger the gaps between the numbers (between 128 and 256, 127 whole numbers are missing). For the table to be of any use, you need almost no gaps between neighbouring numbers – which means choosing a base very close to 1.

That is exactly what Napier did. As the ratio between neighbouring terms he chose the number 0.9999999 (seven nines after the decimal point), and as the first number of the sequence 10,000,000. The second term is 10,000,000 × 0.9999999 = 9,999,999 – just one less. Each further step takes away only a tiny bit, so the table covers the numbers densely, with practically no gaps.

How he worked it out without computing 10 million numbers

If Napier had wanted to compute the logarithm of every number from 1 to 10,000,000 this way, he would have had to carry out millions of multiplications by hand – one human lifetime would not have been enough. So he used a cleverer method: instead of computing one giant table, he computed three shorter “ladders” of numbers nested inside each other.

The first ladder had 101 numbers and used the ratio 0.9999999 (the one from the previous section). The second ladder had 51 numbers and a slightly bigger step, derived from the end of the first ladder. The third ladder was a table of 69 rows and 21 columns, with an even bigger step. By combining these three ladders and estimating between them, Napier could determine the logarithm of practically any number without computing all 10 million values one by one.

Even so, it took him about twenty years. He described the result in 1614 in the book Mirifici Logarithmorum Canonis Descriptio (A Description of the Wonderful Rule of Logarithms) – a book that was later said to have “freed the world from the burden of calculation”.

Napier’s exact method is mathematically more involved; this is a simplified but factually correct description of the principle. Source: D. Roegel, Napier’s ideal construction of the logarithms, 2012.