It's possible to collect together all of the elements in a finite field of size $p$, $p$ prime, that have multiplicative order $\le \log{(p)}$, and put them in a set $s$. My question is, how often is the size of this set, $|S| \ge \log{(p)}$?
I found this PDF that addresses the multiplicative order modulo $p$, on average, but I couldn't find out how to address my question from that paper. The rest of the literature seems fairly scarce, so I'm hoping someone else can resolve this.