Confused About Step in Proof of Divergence of $\sum \frac{1}{p}$

Question

I was going through the number theory text by Ireland and Rosen, and was following the proof of the divergence of the sum of reciprocal primes. But I came across a step unclear to me.

The proof so far: https://imgur.com/XbPkBak&JeMOqku

The step itself: https://imgur.com/XbPkBak&JeMOqku#1

I cannot see the equality between the two sides. Help is appreciated.

Answer 1

@user3663006 , the logarithmic power series is

$$\log(1-x)=-\sum_{n=1}^\infty\frac{x^n}n\;\;,\;\;|x|<1$$

and from here

$$\log(1-p^{-s})=-\sum_{n=1}^\infty\frac{p^{-ns}}n\;\ldots$$