How does taking the logarithm of negative numbers compare to taking logarithm of positive numbers?

Question

Is this an accurate statement: The logarithm of negative number is not unique but the logarithm of a positive number is unique. I know the first clause is true but I'm not sure if it applies to positive numbers as well.

Answer 1

The "branch cut" is generally taken to exclude the negative real axis, which is why you may think the log of a negative real number is not unique. You can take whichever branch cut you like.

Answer 2

A positive real number has a unique real logarithm, but infinitely many complex valued logarithms. If $x>0$, then $$ \log x+2\,k\,\pi\,i,\quad k\in\mathbb{Z}, $$ where $\log x$ is the unique real logarithm of $x$, are logarithms of $x$: $$ e^{\log x+2\,k\,\pi\,i}=x,\quad k\in\mathbb{Z}. $$

Answer 3

If $e^a=5$ then $e^{a\pm2\pi i} = 5$ and $e^{a\pm4\pi i} = 5$, etc., so in a sense $a+n\pi i$ for $n\in\mathbb Z$ are all logarithms of $5$. However, with positive numbers the choice $n=0$ is natural; with negative numbers one can say $e^{a\pm\pi i} = -5$ but the choice between $\text{“}+\text{''}$ and $\text{“}-\text{''}$ is arbitrary.

Just how to define "natural" in this context is an interesting question. If there's an appropriate way to define it, maybe one could prove it's natural, and then we could argue about whether the definition is the “right” one.