## How to Use Log Tables

In previous posts, I pointed out how logarithms took the hardship out of arithmetic in the days before electronic calculators. However, the use of logs hinged crucially on access to an accurate and usable set of logarithm tables. This is probably why logarithms make their appearance only after Gutenburg’s invention of the printing press in Europe. This invention meant that useful log tables could (eventually) be printed cheaply and in bulk, and so instead of being just another special functions, logarithms found a place in the bookshelves of mathematicians, engineers and scientists across the world.

But how did people use these tables? It’s actually a small skill in and of itself. In this post, I’ll outline how log tables were used for most of the 20th Century, having by then evolved into their most streamlined form. I’ll use a set of four figure tables for these examples to keep things simple, but tables with higher precision were available, along with more sophisticated correction methods. This will probably only serve as a refresher for log table veterans.

We’ll stick with the basics for now. Below is a twin set of four-figure log and anti-log tables, in the common base of $10$, listing values $\log_{10}x$ for $x$ from $1.0$ to $10$ and $10^m$ for $m$ from $0.0$ to $1.0$. The values are given to 4 places of decimals, hence the term, four-figure tables. Each table consists of two pages, lying facing on another.

A complete set of 4-figure log tables, on two facing pages.

The corresponding set of 4-figure anti-log or $10^m$ tables.

So, how were these tables read. Let’s focus on the log tables for now. Consider the following crop of the first page of the $\log$ table, showing the top left of the first page.

A section of the 4-figure log tables.

On the leftmost column you can see values of $x$ being listed, in differences of $0.1$, from $1.0$ down to $2.6$. On the top row, additional second place digits for $x$ are listed, along with another list to the right for third place digits — but more on these later.

Returning to the values of $x$ in the first column, the second column lists the corresponding values of $\log_{10} x$, minus a leading zero to the left of the decimal point. So for example: To calculate the value of $\log_{10}(1.4)$, simply travel down the leftmost column to $1.4$ and read off the digits next to this to obtain $\log_{10}(1.4)=0.1461$, which is the value correct to 4 decimal places. The image below illustrates this process, starting from the solid red block.

Looking up $\log_{10} 1.4= 0.1461$.

## Logs make an Impact: Kepler’s Third Law

Johannes Kepler. Discoverer of the laws of planetary motion, and early logarithm enthusiast.

It was only a few short years after Napier’s discovery of logarithms that they made their first, and lasting, impact on the sciences. They were instrumental in uncovering the most important of Kepler’s Laws of planetary motion.

The German astronomer Johannes Kepler had published his first two laws of planetary motion in 1609. These were

• I — The planets orbit the sun in ellipses, with the sun at one focus.
• II — The line joining a planet to the sun sweeps out equal areas in equal times.

But for nearly a decade afterwards Kepler, a prodigious calculator, laboured in vain to find some further relationship hidden in the masses of observational data that the late Tycho Brahe has gathered. In particular, he had yet to find a law linking the orbits of all the planets in some way.

Enter the logarithm. Kepler was an instant convert on discovering Napier’s logs in 1616, and would eventually publish tables of his own. But before this, he seems to have used logs to finally discover the elusive third law he’d been looking for.

As in the previous post, we’ll ‘cheat’ a little here by using modern notation (and data). Kepler had data on the orbital parameters of the first six planets, among them the orbital period $T$, and the semi-major axis $R$. Below is a table of suitably normalised values of $T$, $R$ for each of the planets in the solar system, and their logarithms $\log_{10}(T)$ and $\log_{10}(R)$. To make things clearer here, two planets which were as yet undiscovered in Kepler’s time — Uranus and Neptune — have both been included in the data.
$\begin{array}{l|c|c|c|c|} \text{Planet} & \text{T (/yrs)}& \text{R (/10^6 km)} & \log_{10}(T) & \log_{10}(R)\\ \hline \text{Mercury (m)} & 0.24 & 57.91 &-0.618 & 1.763\\ \text{Venus (V)} & 0.62 & 108.21 &-0.211 & 2.034\\ \text{Earth (E)} & 1.0 & 149.6 &0.0 & 2.175\\ \text{Mars (M)} & 1.88 & 227.94 & 0.274 & 2.358\\ \text{Jupiter (J)} & 11.86 & 778.34 &1.074 & 2.891\\ \text{Saturn (S)} & 29.45 & 1426.71 & 1.469 & 3.154\\ \text{Uranus* (U)} & 84.02 & 2870.63 &1.924 & 3.458\\ \text{Neptune* (N)} & 164.79 & 4498.39 &2.217 & 3.653\\ \hline \end{array}$

Plotting these logarithm pairs gives what is called a log-log plot of the data, and this is also shown below.

Log-Log plot of the orbital period $T$ and semi-major axis $R$ of eight planets in the solar system. The graph is a perfect straight line, revealing a power law relationship between the variables.

## Napier’s Discovery of Logs

John Napier, the inventor of the first Log tables.

Logarithms were discovered by the Scottish mathematician John Napier in the early 1600′s. Napier was looking for ways to speed up arithmetic, and also developed what are now called Napier’s bones towards the same purpose. But we’ll focus on logs in this post.

Napier’s method of deriving logs is a little confusing. Napier came at the problem by considering the motion of two particles, one moving with constant velocity, and another whose motion slowed as it approached its destination. This was all in the days before calculus, so Napier had to explain how to relate quite complicated motions using only static or geometrical arguments. This is probably where a lot of the historical confusion comes from. Here, I’m just going to ‘cheat’ by using calculus to explain the basic ideas.

Napier’s two particles are linked by their motion in time. One moves to the right with constant speed $u$, travelling a distance $d=ut$ over time. Napier called this “arithmetic” motion. The second particle starts at one end of an interval of length $A$, and `falls’ back towards the other end $O$, but with a speed $v$ which is proportional to its current distance from $O$. Napier called this “geometric” motion.

Let’s ‘cheat’ and use calculus to try and understand this second motion. Let $s$ be the current distance of the particle from $O$. Then the proportional velocity is given by $v=-\alpha s$, for some constant $\alpha$. And since the particle starts at $s=A$, we just need to solve the differential equation
$v=\frac{ds}{dt}= -\alpha s ; \qquad s(0)=A$
which has solution
$s=Ae^{-\alpha t}, \qquad \text{or} \qquad s=A\beta^{-t}$

So the first thing we can see is that the particle never reaches $O$. It ‘falls’ to the left forever. Secondly, we can see that the particle reduces its current distance to $O$ by a fixed proportion $\beta$ with each time step; $s(1)=A\beta$, $s(2)=A\beta^2$, $s(3)=A\beta^3$ and so on. Distance is behaving like a geometric sequence $ar^n$, hence the term “geometric motion”.

## The Arithmetic of Logs

Logarithms are as a good place to start as any to start a maths blog.

Before everyone calculated using one of these…

As they progress in mathematics, everyone eventually encounters logarithms. Nowadays these are introduced as a purely algebraic tool, useful for bringing variables down from powers and for solving index equations. But not so long ago logarithms were primarily an arithmetical tool, indispensable in basic calculation. Via tables or slide rules, they were used daily by millions to multiply and divide, and to take powers and roots. You don’t even have to be that old to actually remember doing this.

However, logarithms in basic arithmetic were made obsolete by the arrival of affordable electronic calculators in the late 1970′s. The world collectively put away its slide rules and log tables, and nowadays most people — myself included — will study the complete algebra of logs without ever learning about their original purpose and function. A little sad when you think about it.

So, this post is intended as an introduction to the basic arithmetic of logs for those still unaware of it; and as a refresher for older/enlightened readers. However, I’ll also show how logs live on as alternative to direct calculation. Even today, when the chips are down, the power of logs can still trump direct calculation by even the most powerful computers.

## The log rules and the log tables

Most learn the algebra of logs from their definition as an inverse of exponentiation. If $b^m=x$, then $\log_b x = m$, and from this definition the basic rules of logs follow. Lists vary, but these rules usually include the following.
\begin{align}
\bullet&\ \text{I} &\log_b(xy)&= \log_b(x) + \log_b(y)\\
\bullet&\ \text{II} &\log_b(x/y)&= \log_b(x) – \log_b(y)\\
\bullet&\ \text{III}&\log_b(x^p)&= p\log_b(x)\\
\bullet&\ \text{IV} &\log_b(a)&= \frac{\log_c(a)}{\log_b(c)}
\end{align}

This is essentially as much as modern students learn (probably wondering all the while what on earth they’re doing). But originally, these rules were always paired with a set of log and anti-log tables. It was the combination of the rules and the tables which made logs so useful in the first place.

…they multiplied and divided using one of these.

Two sets of tables were needed: A log table, listing out values of $\log_b(x)$ for a given base $b$; and also an anti-log table, which listed out values of $b^m$ in the same base. For practical purposes, the base was almost always 10. Logs were given for values of $x$ from 1.0 up to 10, while anti-logs were given for values of $m$ from 0.0 to 1. Real log tables had far more elaborate designs which I’ll discuss in a later post, but for now, we’ll consider them to be simple lists of values like so
$\begin{array}{c|c} x & \log_{10}(x)\\ \hline 1.0 & 0.0000\\ 1.1 & 0.0414\\ 1.2 & 0.0792\\ 1.3 & 0.1139\\ 1.4 & 0.1461\\ 1.5 & 0.1761\\ 1.6 & 0.2041\\ 1.7 & 0.2304\\ 1.8 & 0.2553\\ 1.9 & 0.2788\\ 2.0 & 0.3010\\ 2.1 & 0.3222\\ \vdots & \vdots\\ 10.0 &1.0000 \end{array} \qquad \begin{array}{c|c} m & 10^m\\ \hline 0.00 & 1.000\\ 0.01 & 1.023\\ 0.02 & 1.047\\ 0.03 & 1.072\\ 0.04 & 1.096\\ 0.05 & 1.122\\ 0.06 & 1.148\\ 0.07 & 1.175\\ 0.08 & 1.202\\ 0.09 & 1.230\\ 0.10 & 1.259\\ 0.11 & 1.288\\ \vdots & \vdots\\ 1.00 &10.000 \end{array}$

Depending on how large you were prepared to make your tables (read. how much time and money you were prepared to spend), your tables could have finer increments for $x$ and $m$, give more decimal places for the logs and anti-logs, and supply extra corrections for additional accuracy. You could also dispense with tables altogether and use slide rules — devices which became an art in and of themselves.

We’ll just consider tables for now, and how they were used with each of the rules above. In what follows below, it will be assumed that we have access to a more finely graded set of tables such as

$\begin{array}{c|c} x & \log_{10}(x)\\ \hline 1.000 & 0.0000\\ 1.001 & 0.0004\\ 1.002 & 0.0009\\ 1.003 & 0.0013\\ \vdots & \vdots \end{array} \qquad \begin{array}{c|c} m & 10^m\\ \hline 0.0000 & 1.0000\\ 0.0001 & 1.0002\\ 0.0002 & 1.0005\\ 0.0003 & 1.0007\\ \vdots & \vdots \end{array}$

## I: Logs turn multiplication into addition (plus lookups)

Now comes the power of logs. First we must cast our mind back to the days before calculators (There will be a payoff for the modern age later on). Suppose you needed to calculate the product of $4.89 \times 3.37$ to a relatively good approximation. You could just multiply things out, but that would take a bit and you’d need to be quick and accurate with your times tables.

Log tables offered a faster option. A quick lookup in the log table reveals that $\log_{10}(4.89)=0.6893$ and $\log_{10}(3.37)=0.5276$. Now, consider taking the log of the unknown product to the base ten, and applying rule I.
\begin{align*}
\log_{10}\brkt{4.89 \times 3.37} &= \log_{10}(4.89)+\log_{10}(3.37)\\
&\cong 0.6893+0.5276
\end{align*}
So it follows that approximately,
$\log_{10}\brkt{4.89 \times 3.37} \cong 1.2169$
We still don’t know what the product is, but taking the exponent of both sides and manipulating indices gives.
$4.89 \times 3.37 \cong 10^{1.2169} = 10^{0.2169} \times 10^1$
So the product is an integer power of $10$, times the mantissa $10^{0.2169}$. Looking up $10^m$ in the anti-log tables gives $10^{0.2169} \cong 1.647$ and so
$4.89 \times 3.37 \cong 1.647 \times 10^1= 16.47$
$4.89 \times 3.37 = 16.4793$