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We construct log-motivic cohomology groups for semistable varieties and studythe $p$-adic deformation theory of log-motivic cohomology classes. Our mainresult is the deformational part of a $p$-adic variational Hodge conjecture forvarieties with semistable reduction: a rational log-motivic cohomology class inbidegree $(2n,n)$ lifts to a continuous pro-class if and only if its Hyodo-Katoclass lies in the $n$-th step of the Hodge filtration. This generalises atheorem 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 thelogarithmic Picard group. Along the way, we relate log-motivic cohomology tologarithmic Milnor $K$-theory and the logarithmic Hyodo-Kato Hodge-Wittsheaves.

A log-motivic cohomology for semistable varieties and its $p$-adic deformation theory