Series for log(Riemann zeta) in terms of Mangoldt function

The following formula holds: $$\log\left( \zeta(z) \right)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\Lambda(k)}{\log(k) k^z},$$ where $\log$ denotes the logarithm and $\zeta$ denotes the Riemann zeta.