Write the powers of 2 in a row and, below them, the exponents:

$$\begin{array}{c|ccccccccccc} 2^n & 1 & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512 & 1024\\ \hline n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \end{array}$$

The top row is a geometric progression: each number is the previous one times 2. The bottom row is an arithmetic progression: each number is the previous one plus 1. Now try to multiply $16\times64$. Under 16 is 4; under 64, 6. Add them: $4+6=10$. Above 10 is 1024. And indeed

$$16\times64=2^4\times2^6=2^{4+6}=2^{10}=1024.$$

Multiplying above is adding below. It is the rule $2^a\times2^b=2^{a+b}$, and it is the whole idea of logarithms.

An old idea

Archimedes had already noticed the rule in The Sand Reckoner, around 250 BC, when he invented a system for writing enormous numbers. In 1544 the German Michael Stifel, in his Complete Arithmetic, put the two progressions side by side, exactly as above, and wrote that addition in the arithmetic progression corresponds to multiplication in the geometric one, subtraction to division, and halving to taking the square root.

Above (geometric) Below (arithmetic) Example
multiply add $32\times8=256$ because $5+3=8$
divide subtract $1024\div16=64$ because $10-4=6$
square double $32^2=1024$ because $2\times5=10$
take the square root halve $\sqrt{256}=16$ because $8\div2=4$

The gaps

If the idea was so old, why did nobody use it before Napier? Because the table has gaps. It works for $16\times64$ but not for $17\times63$: neither 17 nor 63 appears in the top row. Between 16 and 32 there is nothing, and between 512 and 1024 the gap is even bigger. A useful table needs practically every number in its top row.

Napier’s solution was to replace 2 by a number very close to 1. The powers of 2 grow quickly and leave huge gaps; the powers of $1.0001$, for example, grow very slowly:

$$1.0001^{1}=1.0001,\quad1.0001^{2}=1.00020001,\quad\dots,\quad1.0001^{6932}\approx2.0000.$$

With such a small step, the top row passes close to any number, and the bottom row gives its “exponent”. Napier used a number even closer to 1, $1-10^{-7}=0.9999999$, and spent years computing its powers. The result was the first table of logarithms.

Exercise: use the table of powers of 2 to compute 64 × 256 and 4096 ÷ 32.

$64=2^6$ and $256=2^8$, so $64\times256=2^{6+8}=2^{14}=16{,}384$. For the division, $4096=2^{12}$ and $32=2^5$, so $4096\div32=2^{12-5}=2^7=128$.

Exercise: in the progression of powers of 1.0001, what is roughly the exponent of 3, given that that of 2 is 6932?

We do not need to compute again. Since $1.0001^{6932}\approx2$, the exponent of any number is proportional to its logarithm: that of 3 is about $6932\times\frac{\ln3}{\ln2}=6932\times1.585\approx10{,}987$. Indeed, $1.0001^{10{,}987}\approx3.000$. The ratio between exponents does not depend on the step chosen; this is what guarantees that all tables of logarithms are, at bottom, the same table with different scales.

Sources: John Napier, Mirifici logarithmorum canonis descriptio (Edinburgh, 1614) · Michael Stifel, Arithmetica integra (Nuremberg, 1544) · Julian Havil, John Napier: Life, Logarithms, and Legacy (2014).

All topics on John Napier · ← The life of John Napier · Napier’s two points: the definition of the logarithm →

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