Convergence of the sum $\sum\limits_{p}^{}\sum\limits_{k=1}^{\infty}\frac{\log p}{p^{ks}}$

Question

How can I prove the following sum converges, where $s>1$ and the sum is over all primes. $$\displaystyle\sum_{p}^{}\displaystyle\sum_{k=1}^{\infty}\frac{\log p}{p^{ks}}$$ I've tried grouping terms in various ways, but it seems difficult to prove, because of being over all primes and all powers. More elementary methods would also be appreciated, without assuming any facts about the Riemann zeta function