Bound on a sum over primes $\sum \frac{\log p}{p}$

Question

I would like to prove the following $$ \sum_{w \leq p \leq z} \frac{\log p}{p} - \log (z/w) \leq C $$ where the sum is over the primes $p$ and $C>0$ is some constant. (and $w \geq 2$). I have tried using partial summation with prime number theorem, but I haven't been able to get the bound.. Any comments would be appreciated. Thank you!