  We construct log-motivic cohomology groups for semistable varieties and study
the $p$-adic deformation theory of log-motivic cohomology classes. Our main
result is the deformational part of a $p$-adic variational Hodge conjecture for
varieties with semistable reduction: a rational log-motivic cohomology class in
bidegree $(2n,n)$ lifts to a continuous pro-class if and only if its Hyodo-Kato
class lies in the $n$-th step of the Hodge filtration. This generalises a
theorem of Bloch-Esnault-Kerz which treats the good reduction case. In the case
$n=1$ the lifting criterion is the one obtained by Yamashita for the
logarithmic Picard group. Along the way, we relate log-motivic cohomology to
logarithmic Milnor $K$-theory and the logarithmic Hyodo-Kato Hodge-Witt
sheaves.
