= Solution
Near zero, the absolute value of the integrand is comparable to
$$
t^{z-1}|\log t|,
$$
which is integrable for $z>0$. At infinity, the factor $e^{-t}$ dominates every power of $t$ and $\log t$, so the integral also converges there.
Direct use of Laplace's method is inconvenient because the phase
$$
z\log t-t
$$
does not have the form $z\phi(t)$ with a fixed phase: its stationary point $t=z$ moves to infinity as $z\to\infty$. Rescaling is needed to fix the maximum.
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