Here is a question from a 1952 O-level exam paper:

Calculate, correct to three significant figures, \[\frac{1{\cdot}152\times(3{\cdot}902)^3}{(5{\cdot}463)^2} .\]

This is relatively straightforward to do using a calculator.

But electronic calculators only became commonly available from the late 1970s, so how did students of the 1950s answer a question like this?

They used logarithm tables (usually called “log tables”) where we would now use calculators, and in this resource we will explore how logarithms can help us to do calculations.

We’ll start with something simpler, and look at the exam question in a later section.

Use logarithm tables to evaluate, correct to three significant figures, \[336\times6290 .\]

We will do this in a series of steps.

Can you write the numbers in this calculation (\(336\) and \(6290\)) in standard form?

In this video, you can see someone working out \(\log 336\) using log tables.

Can you describe what they are doing and explain how it relates to standard form?

The logarithm is found in two parts.

The table gives the digits of \(\log{3{\cdot}36}\) (namely \(5263\)) and these go after the decimal point.

To get from \(3{\cdot}36\) to \(336\) we have to multiply by \(10^2\), so the whole number part of the logarithm is \(2\).

You can think of it as
\[\begin{align*}
336&=3{\cdot}36\times10^2\\
\log336&=\log{(3{\cdot}36\times10^2)}\\
&=\log{3{\cdot}36}+2\\
&=2+0{\cdot}5263
\end{align*}\]

Can you work out \(\log 6290\) using the log tables?

Check your answer in the following video:

This clip shows how we can combine the two logs to work out the log of the answer.

(If you have already learnt about the laws of logarithms, you might like to try to do this step yourself before watching the video clip.)

One of the laws of logarithms states that \[\log ab = \log a + \log b\] so by adding the two logarithms we have found the logarithm of the answer.