Let k be a perfect field of characteristic p>0, and let X be a proper scheme over W(k) with semistable reduction. I shall formulate an analogue of the Fontaine-Messing variational p-adic Hodge conjecture in this setting. To get there, I shall define a logarithmic version of motivic cohomology for the special fibre X_k. This theory is related to relative log-Milnor K-theory, logarithmic Hyodo-Kato Hodge-Witt cohomology, and log K-theory. With this in hand, I shall prove the deformational part of the conjecture, simultaneously generalising the semistable p-adic Lefschetz (1,1) theorem of Yamashita (the case r=1) and the deformational p-adic Hodge conjecture of Bloch-Esnault-Kerz (the good reduction case). This is joint work with Andreas Langer.