1.3 Multiplication and Division

Suppose we have two numbers, \(x\) and \(y\) and we wish to multiply them together. If

\[ x = 10^p ~~ {\rm and} ~~ y = 10^q, ~~~~ {\rm then} ~~~~ x\times y = 10^p \times 10^q = 10^{p+q}. \] Thus we see that the logarithm of \(x\times y\) will be the sum of the exponents \(p\) and \(q\), which are themselves the logarithms of the original numbers. This gives the very important result:

\[ \log (x\times y) = \log x + \log y. \] Adding the logarithms of two numbers gives the logarithm of the product of the two numbers!

As for division, just remember that dividing by a number is equal to multiplying by that number’s reciprocal:

\[ x \div y = \frac{x}{y} = x\times \frac{1}{y} = 10^p\times \frac{1}{10^q} = 10^p\times 10^{-q} = 10^{p-q}. \] So, subtracting the logarithms of two numbers gives the logarithm of their ratio:

\[ \log \left( \frac{x}{y} \right) = \log x - \log y. \]

The basic operations of the slide rule involve adding and subtracting distances along the rule which are proportional to the logarithms of the numbers on the scales.

Looking at our previous plot, we can see that adding the logarithms of two numbers, let’s say 2.5 and 6, gives us the logarithm of their product – in this case, 15: