Integral representation of the mean value of the partition function logarithm, i.e., $\langle \log Z \rangle $

Question

I'm reading an old paper and it presents some sort of identity but no reference. It says the following: In order to calculate the average of $\log Z$ (here $Z$ is the canonical partition function), one can use an integral representation of the logarithm: \begin{equation} \langle \log Z \rangle = \int_0^{\infty} \frac{e^{-t}- \langle e^{-tZ} \rangle }{t} dt \end{equation} I have no clue on how to prove or, at least, understand this identity. So please help me understand this.