For log smooth rigid analytic varieties, Kummer pro-étale B_dR cohomology is isomorphic to log de Rham cohomology, and logarithmic B_dR^+ cohomology gives degeneration of Hodge-Tate and Hodge-log de Rham spectral sequences when proper.
Notes on crystalline cohomology
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On the Kummer pro-\'etale cohomology of $\mathbb B_{\operatorname{dR}}$
For log smooth rigid analytic varieties, Kummer pro-étale B_dR cohomology is isomorphic to log de Rham cohomology, and logarithmic B_dR^+ cohomology gives degeneration of Hodge-Tate and Hodge-log de Rham spectral sequences when proper.