Proof that $\lim_{s\to 1}\ln \zeta (s)$ diverges

Question

I'm trying to prove that $$\displaystyle\lim_{s\to 1}\ln\zeta (s) \,\text{diverges}. \tag*{(*)}$$ Initially, I thought of $$\displaystyle\lim_{s\to 1}\ln\zeta (s)=\ln\displaystyle\lim_{s\to 1}\zeta (s)\implies \left(\displaystyle\lim_{s\to 1}\ln \zeta (s) \,\text{converges}\,\iff \ln\lim_{s\to 1}\zeta (s)\,\text{converges}\right).$$ But then we have that it's not always true that $$\displaystyle\lim_{s\to a}\ln f(s)=\ln \displaystyle\lim_{s\to a} f(s),$$ so I ruled the "proof" out. How can I prove $(*)$?