Log blow-ups and global resolutions in rigid geometry

In this article, we study log blow-ups in the setting of log adic spaces. As in the theory of log schemes, we prove the existence of global resolutions of toric singularities for fs log adic spaces log smooth over a $p$-adic field, and the acyclicity of certain sheaves under log blow-ups. One new feature of (log) rigid analytic varieties is their integral structure sheaf: we prove that it is also almost acyclic under log blow-ups in the Kummer étale topology. We also supply some foundational results on log adic spaces.