Question about the $\ln$ of a product of primes

Question

For what conditions of $s\in\mathbb{C}$ does this hold?

1. $$\ln\left[\prod_{\text{p}\in\mathbb{P}}\frac{1}{1-\text{p}^{-s}}\right]=\sum_{\text{p}\in\mathbb{P}}\ln\left[\frac{1}{1-\text{p}^{-s}}\right]$$
2. $$\ln\left[\frac{1}{1-\text{p}^{-s}}\right]=-\ln\left(1-\text{p}^{-s}\right)$$

I dont know where to start.

But what I did was:

• $$\text{p}^s=\text{p}^{\Re(s)+\Im(s)i}=\text{p}^{\Re(s)}\cos\left(\Im(s)\ln(\text{p})\right)+\text{p}^{\Re(s)}\sin\left(\Im(s)\ln(\text{p})\right)i$$
• $$\text{p}^{-s}=\frac{1}{\text{p}^s}=\frac{1}{\text{p}^{\Re(s)+\Im(s)i}}=\text{p}^{-\Re(s)}\cos\left(\Im(s)\ln(\text{p})\right)+\text{p}^{-\Re(s)}\sin\left(\Im(s)\ln(\text{p})\right)i$$