Evaluate the complex logarithms $\log(i)$ and $\log(3+4i)$

Question

How to evaluate $$\log(i)?$$
Also, how to evaluate $$\log(3+4i)?$$
I am reading complex analysis and I know that logarithm is a multibranched function and is periodic. What I have for the definition is $$\log(z) = \log(\rho)+i(\theta +2k\pi).$$
That is for when $z$ is complex.
I also have, $\log(z)=\log|z|+i(\arg(z)+2k\pi)$
I don't know where to start, so please write a detailed and step-by-step solution if you can, thank you very much!

Answer 1

The idea is that:

$$\log(z) = \log(\rho)+i(\theta +2k\pi),$$

where $z = \rho e^{i\theta}.$

When $z = i$, then $z = e^{i\frac{pi}{2}}$. Hence, $\rho = 1$ and $\theta = \frac{\pi}{2}$. Finally:

$$\log(i) = \log(1) + i \left(\frac{\pi}{2} + 2k \pi\right) = \frac{i\pi}{2}(1 + 4k). $$

When $z = 3+4i$, we have that $\rho = \sqrt{3^2 + 4^2} = 5$ and $\theta = \arctan \left(\frac{4}{3}\right).$

Hence:

$$\log(3+4i) = \log(5) + i \left(\arctan \left(\frac{4}{3}\right) + 2k \pi\right). $$