Let $(s,x) \mapsto \gamma(s,x) := \int_2^x (\log z)^{-s}\mathrm{d}z$ be the incomplete gamma function (I'm not sure about this naming). It is well-known that for large $x \gt 0$, the following asymptotic estimate is valid
$$ \gamma(1,x) = O(x/\log x). $$
Question. For large $x \gt 0$ and $s \gt 0$, what is an good asymptotic estimate for the integral $\gamma(s,x)$ ?