A sort of continuation of Comparing distributions with moments

Suppose I have some estimates of the moments of a non-negative random variable $X$: $$\log \mathbb{E}(X^n) = n \log n + (\beta-1)n + O(\log n),$$ with $\beta > 0$. Can I conclude that the tail $$\mathbb{P}(X>x) \sim A e^{-c x}$$ for some $A$ and $c$?