When can I assume $\operatorname{Log}$ means the natural log?

Question

I have a problem that says:

Express $e^{\operatorname{Log}(3+4i)}$ in the form $x+iy$.

However, in class we usually only deal with natural log (this problem is from a textbook). How do I know when I can interpret ''Log'' as the natural log?

Answer 1

Since $z\mapsto e^z$ is periodic with period $2\pi i$, it is not one-to-one, and so its inverse is a "multiple-valued function" (and so not really a "function" as textbooks define that term over the past hundred years or so). Thus $$ \operatorname{Log}(3+4i) = \log\sqrt{3^2+4^2} + i (\theta_0 + 2\pi n) $$ and $\theta_0 + 2\pi n$ is any of the angels in the first quadrant whose tangent is $4/3$.

Exponentiating this is exponentiating any of the complex number that when exponentiated yield $3+4i$. Hence $$ e^{\operatorname{Log}(3+4i)} = 3+4i. $$