The $r^{th}$ moment of the divisor function, for $r\geq 1,$ is well known to obey

$$ \sum_{n\leq x} \tau(n)^r\sim C_r x (\log x)^{2^r-1} $$. where $C_1=1.$ In a paper by Florian Luca and L. Toth, available at https://arxiv.org/abs/1703.08785, the constant $C_r$ is also given.

What about general $r \in (1,\infty)$? Does this expression still provide a good order of magnitude approximation?