  We prove that if a nonautonomous system has in a certain sense a fast
convergence to equilibrium (faster than any power law behavior) then the time
$\tau _{r}(x,y)$ needed for a typical point $x$ to enter for the first time in
a ball $B(y,r)$ centered in $y$, with small radius \ $r $ scales as the local
dimension of the equilibrium measure \ $\mu $ at $y$, i.e. $$
\underset{r\rightarrow 0}{\lim }\frac{\log \tau _{r}(x,y)}{-\log r}% =d_{\mu
}(y).$$
  We then apply the general result to concrete systems of different kind,
showing such a logarithm law for asymptotically authonomous solenoidal maps and
mean field coupled expanding maps.
