How to expand and compute log(a + b)? [closed]

Question

I would like to know the complete expansion of log(a + b).

For example

log(a * b) = log(a) + log(b);
log(a / b) = log(a) - log(b);

Similar to this, is there any expansion for log(a + b)?

Answer 1

Why would you ever want to do this? The property that log (a*b) = log a + log b is only useful because it transforms a multiplication operation into an addition operation. log (a+b) already involves only an addition, so it makes no sense to have any further expansion.

Of course you can always use one of the several series for computing logarithms, but the fastest way would be to simply compute log (a+b) directly. For that matter, on most computers, even log (a*b) is going to be faster than log a + log b, since the latter involves an extra logarithm operation.

Answer 2

In general, one doesn't expand out log(a + b); you just deal with it as is. That said, there are occasionally circumstances where it makes sense to use the following identity:

log(a + b) = log(a * (1 + b/a)) = log a + log(1 + b/a)

(In fact, this identity is often used when implementing log in math libraries).