It is often useful to combine several different logs, being added or subtracted, into a single log that can then be manipulated more easily.

For example it is easier to find log(6) than log(3)+log(4) – log(2)

In order to combine logs like this you need to use the following rules:
log_c(a) + log_c(b) = log(ab)
and
log_c(a)-log_c(b) = log(a/b)
Remember: all logs must be in the same base, you cant use this to add log₂(a) + log₃b

This rule can be proved quite simply as follows:

Let all logs be to the base c.

if log a = m and log b = n
then a=c^m and b = cⁿ
so ab = c^mcⁿ= c^m+n
by taking logs we obtain
log(ab)=log(c^m+n) = m + n
and by substituting in the values of m and n we get the required result
log(ab) = log a + log b

The result for log (a/b) can also be found in this way but I’ll leave that one for you to work out.

By David Woodford