  Let $R$ be a strictly local ring complete for a discrete valuation, with
fraction field $K$ and residue field of characteristic $p > 0$. Let $X$ be a
smooth, proper variety over $K$. Nicaise conjectured that the rational volume
of $X$ is equal to the trace of the tame monodromy operator on $\ell$-adic
cohomology if $X$ is cohomologically tame. He proved this equality if $X$ is a
curve. We study his conjecture from the point of view of logarithmic geometry,
and prove it for a class of varieties in any dimension: those having
logarithmic good reduction.
