  We examine the conjecture, due to Champanerkar, Kofman, and Purcell that
$\text{vol}(K) < 2 \pi \log \det (K)$ for alternating hyperbolic links, where
$\text{vol}(K) = \text{vol}(S^3\backslash K)$ is the hyperbolic volume and
$\det(K)$ is the determinant of $K$. We prove that the conjecture holds for
2-bridge links, alternating 3-braids, and various other infinite families. We
show the conjecture holds for highly twisted links and quantify this by showing
the conjecture holds when the crossing number of $K$ exceeds some function of
the twist number of $K$.
