TITLE: what is the application of log(x) where x is negative number
QUESTION [2 upvotes]: what is the application of log(x) where x is negative number?
Anyone knows real usecase?

REPLY [2 votes]: Logarithm is normally defined only on positive reals. Logarithm of $0$ seems to make little sense, and there is no natural choice for the negative numbers. Note that the minimal requirement for $\log$ is that $\exp(\log(x)) = x$, so for $x < 0$ you need to take $$\log(x) = \log(|x|) + \pi i + 2k \pi i,\qquad k \in \mathbb{N}$$ where the terms correspond (consecutively) to the right absolute value, the minus sign, and the fact that $\exp$ is periodic with period $2 \pi i$ if you onsider it as a function on the complex plane (if you wanted to stick to reals, you have no way of making $\exp(y)$ negative). You can, by convention, take some fixed value of $k$ (say $k = 0,-1,...$), but no such choice is canonical. 
A fact that might interest you is that if you removed a halfline starting from $0$ in the complex place (say, different than $\mathbb{R}_+$), then on the remainder there would be exactly one possible choice of logarithm so that $\exp \circ \log = \mathrm{id}$.
{"set_name": "stack_exchange", "score": 2, "question_id": 115986}