  The main objects of study are adic spaces with logarithmic structures. After
establishing the basic definitions, we analyze the Kummer \'etale and
pro-Kummer \'etale topologies on log adic spaces. In particular, we show that
log adic spaces are locally "log affinoid perfectoid" in the pro-Kummer \'etale
topology. As an application, we prove finiteness of cohomologies for Kummer
\'etale $\mathbb{F}_p$-local systems on proper log smooth adic spaces.
