  In this paper, we prove a result on the $2$-adic logarithm of the fundamental
unit of the field $\mathbb{Q}(\sqrt[4]{-q}) $, where $q\equiv 3\bmod 4$ is a
prime. When $q\equiv 15\bmod 16$, this result confirms a speculation of
Coates-Li and has consequences for certain Iwasawa modules arising in their
work.
