# 1 Review of the Logarithm

The elementary calculations performed using slide rules utilize the concept of the logarithm, invented in the early 1600’s by John Napier of Scotland.^{1} Napier introduced the name *logarithm* as a combination of two ancient Greek terms *logos*, meaning “divine reason,” and *arithmos*, meaning number. He was looking for a relationship between an arithmetic and a geometric progression, so that tables could be generated that would aid in performing multiplication calculations. At that time, calculations in astronomy and, perhaps most importantly, in navigation required the multiplication of numbers of up to 6-7 digits each.

To illustrate the issue suppose we want to multiply the two numbers 4873 and 382. This can be performed on paper in the following way:

\[ \begin{matrix} ~ & ~ & ~ & {\tiny 2} & {\tiny 2} & ~ & ~ \\ ~ & ~ & ~ & {\tiny 6} & {\tiny 5} & {\tiny 2} & ~ \\ ~ & ~ & ~ & {\tiny 1} & {\tiny 1} & ~ & ~ \\ ~ & ~ &~ & {\bf 4} & {\bf 8} & {\bf 7} & {\bf 3} \\ ~ & ~ & \times & ~ & {\bf 3} & {\bf 8} & {\bf 2} \\ \hline ~ & {\tiny 1} & {\tiny 2} & {\tiny 2} & ~ & ~ & ~ \\ ~ & ~ & ~ & 9 & 7 & 4 & 6 \\ ~ & 3 & 8 & 9 & 8 & 4 & ~\\ 1 & 4 & 6 & 1 & 9 & ~ & ~\\ \hline {\bf 1} & {\bf 8} & {\bf 6} & {\bf 1} & {\bf 4} & {\bf 8} & {\bf 6}\\ \end{matrix} \]

Of course one needs to know the “times tables” very well. And after a few “carries” (indicated above in small type, such as in “8 times 7 is 56, carry the 5”), keeping things lined up and keeping track of what is being added requires discipline and concentration, especially for situations involving many more digits than the above. And the repeated multiplication or division of several multi-digit numbers can become very tedious very quickly. Napier’s concept was to find for each number a corresponding *logarithm* such that multiplying two numbers was reduced to *adding* their logarithms. Adding two numbers, even if the numbers have many digits, is a simpler task and much less error prone.

The concept that was studied and developed by Napier can be illustrated with the following table:

\(p\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

\(x\) | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 |

The top row – the numbers designated “\(p\)” – run sequentially from 1 to 10. The bottom row – designated “\(x\)” – starts with a base number – “2” in this case – and each consecutive number is the previous result multiplied by the base number. The top row is obviously just a list of the first 10 integers; but the second row is called a *geometric sequence*. Now notice the following. If we take two numbers in the \(x\) list and multiply them, the result falls under the *sum* of the corresponding values in the \(p\) list. For instance, take 2 times 8; the result we know is 16. The corresponding values of \(p\) in the table for the numbers 2 and 8 are 1 and 3. We see that the sum of these \(p\) values is 1 + 3 = 4, and the number in the \(x\) row that falls below this result is \(x\) = 16, the result of multiplying our original two numbers. The values of \(p\) are called *logarithms* of the values of \(x\).

Now suppose we wanted to quickly find 8 times 128. We could take out pencil and paper and write

\[ \begin{matrix} ~ & ~ & ~ & ~ & {\tiny 2} &{\tiny 6} & ~ \\ ~ & ~ &~ & {\bf } & {\bf 1} & {\bf 2} & {\bf 8} \\ ~ & ~ & \times & ~ & {\bf } & {\bf } & {\bf 8} \\ \hline ~ & ~ & ~ & 1 & 0 & 2 & 4 \\ \end{matrix} \]

or, from our table above, we could look up the logarithm of 8 which is 3, and the logarithm of 128 which is 7. Adding the logarithms we find 3 + 7 = 10. We then look at the table to find which number \(x\) has a logarithm of \(p\) = 10. The answer found from the table is \(x\) = 1024. Thus, 8 times 128 is equal to 1024.

In our simple example above, the value of \(x\) that can be used from the table is a rather sparse list. The goal of Napier was to find a technique for computing logarithms of other integers in between in order to fill out a complete table for any value of \(x\). From the table it can be surmised, for example, that using 2 as our base, the logarithm \(p\) for the number \(x\) = 7 is somewhere between 2 and 3. If we knew its logarithm more precisely, we could perform similar computations using the number 7.

Napier’s goal was to determine logarithms of the natural numbers for some suitable base number. Through many years of effort he developed ratios of geometric sequences which allowed him to successfully compute tables of numbers that could be added together in order to perform a multiplication or a division. In modern mathematical terms, he was describing the addition and subtraction of *exponents* of a particular *base* number, which in turn, is equivalent to multiplication and division. It is interesting to note that his development of logarithms came at a time before the development of the concept of exponents and exponential notation (\(b^x\), say). Nonetheless, by creating accurate tables of logarithms and producing scales that were proportional to their values, the invention of the slide rule followed within a few short years. Its user was able to perform quick multiplication and division calculations with sufficient accuracy for a wide variety of computational applications.

A nice discussion of Napier’s development of his early tables of logarithms can be found in Denis Roegel’s,

*Napier’s ideal construction of the logarithms*[Research Report], ffinria-00543934f (2010). See https://hal.inria.fr/inria-00543934/en .↩︎