How Did a Book of Numbers Make Multiplying Easy?

The row number of a number in the chart counts how many 2s are multiplied to make it. Row 5 holds 32, because 2 × 2 × 2 × 2 × 2 = 32. In the language of Grade 9’s indices, row n holds $2^n$, and the row number is the index (or exponent). The law you already know, $2^m \times 2^n = 2^{m + n}$, says that multiplying two numbers of the chart means adding their rows.

The other laws of indices give the chart more tricks. Dividing takes 2s away, so it subtracts rows: $2^m \div 2^n = 2^{m - n}$. A square root is the number that, times itself, gives the number back, so it is half the row: $\sqrt{2^{16}} = 2^8$, since $2^8 \times 2^8 = 2^{16}$. A cube root is a third of the row. Long multiplication becomes addition, division becomes subtraction, and the dreaded square and cube roots become halving and dividing by 3.

The German mathematician Michael Stifel printed a table like this in 1544, the numbers $2^n$ in one row and the counts n above them, and pointed out that multiplying numbers in the bottom row is adding the counts above them. It was his word, exponent, that we still use. But his table had a fatal snag.

The chart’s rows are too far apart because each one is double the last. What if each row were only a little bigger than the last? Start at 1 and multiply by 1.1 each time: 1, 1.1, 1.21, 1.331, 1.4641, … The rows creep up instead of leaping, so every number has a row close to it. The rule still holds, because row n is still $1.1^n$ and $1.1^m \times 1.1^n = 1.1^{m + n}$.

Here is what 3 × 7 = 21 gives in each book. The second column says how many rows the book needs just to climb from 1 to 10.

Each row times	Rows from 1 to 10	Rows of 3 and 7	Answer read	Off by
2	about 3.3	2 + 3 = 5	32	52 %
1.1	about 24	12 + 20 = 32	21.11	0.5 %
1.01	about 231	110 + 196 = 306	21.006	0.03 %
1.0001	about 23,027	10,987 + 19,460 = 30,447	21.0005	0.003 %

The closer the ratio is to 1, the closer every answer comes, and the thicker the book. A book with ratio 1.0001 needs 23,027 rows to get from 1 to 10, but no answer read from it can be off by more than about 0.01 %: each number is within half a step of its row, so the row numbers’ sum is out by at most one step, and one step is 1.0001 times, a change of 0.01 %.

Bürgi’s book works, but its row numbers are awkward: 10 is in row 23,027.0022 and 100 in row 46,054.0044. The idea that made the book easy to use came from the English professor Henry Briggs after reading John Napier’s book of 1614 (the story is below): number the rows so that 10 sits in row 1 exactly, 100 in row 2, 1,000 in row 3, and fill in every row in between.

Then the row number of any positive number x is the power you must raise 10 to, to get x. That number is the logarithm of x (to the base 10, the common logarithm), written $\log x$:

$$\log x = n \quad \text{means} \quad 10^n = x.$$

So $\log 1{,}000 = 3$, $\log 10 = 1$ and $\log 1 = 0$. Napier made up the word from Greek logos, ratio, and arithmos, number: a “ratio-number”, counting how many equal ratios (steps of the same times-size) it takes to get from 1 to x.

The logarithm and the power undo each other, the way squaring and square-rooting do: $10^{\log x} = x$ for every positive x, and $\log 10^n = n$ for every n. Each is the inverse of the other. That is all a book of logarithms is: a table of $10^n$ read backwards.

What a row like 0.30103 means

The rows between 1 and 10 have fractional numbers. What could $10^{0.5}$ mean, “10 multiplied by itself half a time”? Grade 10’s indices answer it: two half steps make one step, so $10^{0.5} \times 10^{0.5} = 10^1 = 10$, and $10^{0.5} = \sqrt{10} \approx 3.162$. So $\log 3.162 \approx 0.5$. In the same way $10^{0.25} = \sqrt{\sqrt{10}} \approx 1.778$, and $10^{0.30103} \approx 2$: the row of 2 is 0.30103. A fractional index is exactly the tool that fills the gaps between the whole powers.

Why the book only needs the numbers from 1 to 10

Every positive number can be written as a number from 1 to 10 times a power of 10 (Grade 8’s scientific notation): $3{,}141 = 3.141 \times 10^3$. Multiplying adds rows, so

$$\log 3{,}141 = \log 3.141 + \log 10^3 = 3 + \log 3.141 = 3 + 0.4971 = 3.4971.$$

The whole-number part, 3, is the characteristic: it only says where the decimal point goes, and you can read it from the number of digits. The part after the point, 0.4971, is the mantissa, and it is the same for 3.141, 31.41 and 3,141,000. A number below 1 works the same way with a negative characteristic: $0.03141 = 3.141 \times 10^{-2}$, so $\log 0.03141 = -2 + 0.4971$, which is −1.5029. The old tables kept the mantissa positive and wrote it $\bar{2}.4971$ (“bar two point 4971”), the bar meaning that only the 2 is negative.

So a book of logarithms only lists the mantissas of the numbers from 1 to 10, and that one book serves every number there is. The common school book was a four-figure table (mantissas to 4 decimal places, for numbers of 3 or 4 figures), with a second table, of antilogarithms, for going back from a logarithm to its number.

Using the book

Here are the few lines of a four-figure table we need (each mantissa rounded to 4 places, as the tables printed them):

Number	1.08	1.25	2.86	3.68	3.69	4.37	5.00
Logarithm	0.0334	0.0969	0.4564	0.5658	0.5670	0.6405	0.6990

Here is the proof, using nothing but the laws of indices. Take any two positive numbers a and b, and call their logarithms p and q. That means $a = 10^p$ and $b = 10^q$. Then

$$ab = 10^p \times 10^q = 10^{p + q}, \quad \text{so} \quad \log(ab) = p + q = \log a + \log b.$$

In the same way $a \div b = 10^{p - q}$ and $a^n = (10^p)^n = 10^{np}$, which give the other two laws of logarithms:

$$\log\frac{a}{b} = \log a - \log b, \qquad \log a^n = n \log a.$$

The last one holds for fractional n too, which is why a square root halves the logarithm ($n = \tfrac12$) and a cube root thirds it.

Any base will do

Nothing above needed 10 in particular. The chart on the wall is a table of logarithms to the base 2: the row of x is $\log_2 x$, the power you raise 2 to, to get x. Bürgi’s rows are logarithms to the base 1.0001. All these books are the same book with the rows renumbered. If $2^n = x$, take common logarithms of both sides: $n \log 2 = \log x$, so

$$\log_2 x = \frac{\log x}{\log 2}.$$

This is the change of base rule, $\log_b x = \dfrac{\log x}{\log b}$. Bürgi’s row numbers are the common logarithms multiplied by $1 \div \log 1.0001$, about 23,027: his row for 2 is 23{,}027.0022 × 0.30103, about 6,931.8.

The same step solves any equation with the unknown in the index. Grade 10 solves $2^x = 32$ by making the bases match. Logarithms solve it when they can’t: how many times must Rs 1 double to pass 1 crore?

A book of logarithms is easy to use and terribly hard to make. Where does 0.30103 come from, with no calculator? Try it before reading on: you can find $\log 2$ to two places in your head.

The trick is a power of 2 that is almost a power of 10: $2^{10} = 1{,}024$, very nearly $1{,}000 = 10^3$. So $10 \log 2 \approx 3$, and $\log 2 \approx 0.3$. The true value is 0.30103. Then the laws give more for free: $\log 4 = 2 \log 2 \approx 0.602$, $\log 8 \approx 0.903$, and $\log 5 = \log(10 \div 2) = 1 - \log 2 \approx 0.699$.

Briggs used the same trick (he started from $2^{10} \div 1{,}000 = 1.024$), and another one. Since a square root halves the logarithm, $\log \sqrt{10} = 0.5$, $\log \sqrt{\sqrt{10}} = 0.25$, and so on. Briggs took the square root of 10, then the square root of that, 54 times in a row, the last ones to more than 30 decimal places, by hand. The 54th root is 1.000 000 000 000 000 127 8…, and its logarithm is $1 \div 2^{54}$. Near 1, he noticed, the logarithm is simply proportional to how much the number exceeds 1, and from there he could work out the logarithm of any number. In 1624 his Arithmetica Logarithmica gave the logarithms of 1 to 20,000 and 90,000 to 100,000, to 14 decimal places.

A table needs looking up. In 1620 Edmund Gunter, professor of astronomy at Gresham College in London, put the logarithms on a ruler instead: he marked each number 1, 2, 3, … 10 at a distance from the end equal to its logarithm (on a 25 cm scale, 2 sits 0.30103 × 25 ≈ 7.5 cm along, 3 at 11.9 cm). Distances on that ruler are logarithms, so adding two distances adds two logarithms. Around 1622 the English clergyman William Oughtred slid two such rulers against each other, and the slide rule was born.

The two rulers do exactly what the book does, without the looking up. Sliding puts the length $\log 2$ on D before the length $\log 3$ on C, so the cursor stands at the length $\log 2 + \log 3 = \log 6$, and D is marked 6 there. A scale like this is the graph of $y = \log x$ laid flat along a line: each number at the height its logarithm would have.

That last question shows how people used slide rules: the rule gives about three figures, and the user places the decimal point by a rough estimate. For an engineer designing a bridge or a navigator plotting a course, three figures were usually enough, and a slide rule was far quicker than a book.

Logarithms as a tool for calculating disappeared within a decade. Logarithms themselves did not:

• Every computer stores numbers the way the book did. A computer keeps a decimal number like 0.000123 as a significand times a power (of 2, inside the machine): the power plays the role of the characteristic, and the old word mantissa is still used for the significand, though it means something slightly different there.
• Solving equations with the unknown in the power: how long a loan takes to double, how long until a population or a sum of money passes a target, how old a fossil is from the carbon left in it. Every one of them is solved by taking logarithms, as with the crore above.
• Scales that go up by multiplying: the earthquake magnitude, sound in decibels and acidity as pH all count powers of 10. That is a topic of its own (below).

• Why Is a Magnitude 8 Earthquake Not Just a Bit Bigger Than a 7? Logarithms as a scale for measuring things that range over many powers of 10.
• What Is e, and Why Does Growth Keep Finding It? The other famous base, and how Napier’s logarithms are tied to it.
• What Is the Biggest Number Anyone Has Used? Numbers so big that even their logarithms are too big to write.
• How Did People Multiply Before Calculators? (Grades 6–8) Doubling and halving, the lattice and Napier’s bones: the ways before logarithms.
• How Much Rice Is on the Chessboard? (Grades 6–8) The powers of 2, and how fast they grow.
• How Does a Calculator Find a Square Root? (Computing, Going Further 9–12) What a machine does in place of Briggs’s 54 square roots.
• How Did a Rounding Error Stop a Missile Defense? (Computing, Going Further 9–12) Numbers stored as a significand and a power, and what their rounding can cost.
• Grade 10, Indices: the laws of indices and fractional indices that this page builds on, and Compound Interest.