Is any identity of the form $$ \log x = \int_0^\infty \frac{ds}s e^{-sx}$$ valid for some range of $x \in \mathbb C$?
Perhaps it will be true with $0$ replaced by some $\epsilon>0$, and up to finite terms, but I'm still interested to see whether one can prove it, perhaps with some manipulations involving Gamma function.
This question arises in quantum field theory.