I am interested in knowing about the distribution of exponentially high order elements in $\mathbb{F}_p$. To be precise let $s$ be of the order $\frac{p}{\log^{k}(p)}$ for some fixed $k$ and integer. Given $p$ being an odd prime. I would like to know about the distribution of elements with order higher than $s$. Are there papers dealing with such a question? I could not find any resources regarding this.

Thanks!!