Euler's Formula for Log?

Question

I'm a bit unsure on what to do for the following question:

Find $ Log(6\pi i) $ and $ e^{6\pi i}$

I'm fine with finding $ e^{6\pi i}$ it's $ Log(6\pi i) $ that I'm unsure about. I was thinking of changing it to exponential form but this doesn't seem right? Any help would be really appreciated

Answer 1

1. The $\ln$ function is not well defined in the complex plane, but we 'can say' (by definition): $$\ln\left(\text{a}+\text{b}i\right)=\ln\left|\text{a}+\text{b}i\right|+\arg\left(\text{a}+\text{b}i\right)$$
2. Use Euler's formula: $$e^{\theta i}=\cos\left(\theta\right)+\sin\left(\theta\right)i$$

Answer 2

HINT:

Do you know the definition of the complex $\textrm{Log}$?