Why is $\ln(\zeta(s))\sim \ln\big(\frac{1}{s-1}\big) $ as $s\to 1$

Question

I have shown that $\zeta(s)$ has a simple pole at 1. Now,I need to show that $\ln(\zeta(s))\sim \ln\big(\frac{1}{s-1}\big) $ as $s\to 1$.

I know I can write: $\zeta(s)=\frac{1}{s-1}+\psi(s)$ where $\psi(s)$ is holomorphic. But after that I'm stuck.

Any Hints or Suggestions?