  We consider slice filtrations in logarithmic motivic homotopy theory. Our
main results establish conjectured compatibilities with the Beilinson, BMS, and
HKR filtrations on (topological, log) Hochschild homology and related
invariants. In the case of perfect fields admitting resolution of
singularities, we show that the slice filtration realizes the BMS filtration on
the $p$-completed topological cyclic homology. Furthermore, the motivic trace
map is compatible with the slice and BMS filtrations, yielding a natural
morphism from the motivic slice spectral sequence to the BMS spectral sequence.
Finally, we consider the Kummer \'etale hypersheafification of logarithmic
$K$-theory and show that its very effective slices compute Lichtenbaum \'etale
motivic cohomology.
