Understanding a series representation of the logarithm of the zeta function

Question

I am reading through M. Ram Murty's Problems in Analytic Number Theory and have the following question regarding the first step in his proof of Dirichlet's Theorem.

Given this definition for the zeta function:

$$\zeta(s) = \prod\limits_{p}\left(1 - \dfrac{1}{p^s}\right)^{-1}$$

Murty presents the following argument as the first step:

$$\log\zeta(s) = -\sum\limits_{p}\log\left(1 - \dfrac{1}{p^s}\right) = \sum\limits_{p}\left(\sum\limits_{n=1}^{\infty}\dfrac{1}{np^{ns}}\right)$$

I am not clear how he is able to remove the negative sign and the $\log$ to get to the last result.

I assume that he is using the Euler Product Formula:

$$\sum\limits_{n=1}^{\infty} \dfrac{1}{n^s} = \prod\limits_{p} \dfrac{1}{1 - p^{-s}}$$

But I don't see how this allows him to get to the last expression.

Thanks very much,

-Larry

Answer 1

It is just using the Taylor series of $\log(1+x)$ in the vicinity of $0$