I am try to calculate: $\log(e^{5i})$, but I think I am doing something bad, I suppose that $e^{5i}=\cos(5)+i\sin(5)$, angle is $\tan(5)$ then... $\log(e^{5i})=\log(1)+i(\tan(5))$?, help.
And more, how do I determinate the set $\operatorname{LOG}(e^{5i})$. Is it $\operatorname{LOG}(e^{5i}=\log(1)+i(\tan5+2k\pi)$ to $k\in\mathbb{R}$? mm help.
On
Let $\log(e^{5i})=z$. This means that $e^z=e^{5i}$. Now, if $z=a+ib$ this means: $$ e^{a+ib}=e^{5i} \iff e^ae^{ib}=e^{5i} $$ so we have : $$ e^a=1 \Rightarrow a=0 $$ and $$ b=5+2k\pi $$ because the exponential is a periodic function with period $2i\pi$. This means that his inverse function (the logarithm) is multi-valued, and you can have a single value, or principal value, only if you fix an interval for the argument.
Complex logarithm is not a universally defined function. $$re^{i\theta} = re^{i\theta + 2\pi k i}$$ for any $k \in \mathbb Z$, so one has to specify which $k$ is preferred before using the complex logarithm. Even after specifying $k$, there is always some ray from the origin in the complex plane on which the formulation of $\log$ is undefined. In the case $e^{5i}$, $$\log(e^{5i}) = 5i + 2πki$$ where $k \in \mathbb Z$ as described above.