Heuristic formula for logarithm of the Frobenius morphism

We show that the logarithm ${\log }_q$ of the Frobenius morphism $x\to x^q$ is given by the formula $x\to x\log x$ (the natural logarithm). In particular, it does not depend on $q$. This is the explicit (although heuristical) formula for the operator conjectured by Hilbert whose eigenvalues coincide with the zeroes of the zeta function.