Most permutations power to a cycle of small prime length

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log\log n$ with the prime between $\log n$ and $(\log n)^{\log\log n}$. The proportion of even permutations with this property is at least $1-7/\log\log n$.