  Fix $\alpha>0$, then by Fej\'er's theorem $ (\alpha(\log
n)^{A}\,\mathrm{mod}\,1)_{n\geq1}$ is uniformly distributed if and only if
$A>1$. We sharpen this by showing that all correlation functions, and hence the
gap distribution, are Poissonian provided $A>1$. This is the first example of a
deterministic sequence modulo one whose gap distribution, and all of whose
correlations are proven to be Poissonian. The range of $A$ is optimal and
complements a result of Marklof and Str\"{o}mbergsson who found the limiting
gap distribution of $(\log(n)\, \mathrm{mod}\,1)$, which is necessarily not
Poissonian.
