  We extend the construction of A$_{\rm inf}$-cohomology by
Bhatt-Morrow-Scholze to the context of log $p$-adic formal schemes over a log
perfectoid base. In particular, using coordinates, we prove comparison theorems
between log A$_{\rm inf}$-cohomology with other $p$-adic cohomology theories,
including log de Rham, log (q-)crystalline, log prismatic, and Kummer \'etale
cohomology, as well as the derived A$_{\rm inf}$-cohomology of certain infinite
root stacks. Along the way, we define and give a combinatorial characterization
of a new class of maps between saturated log schemes, called pseudo-saturated
maps, which is of independent interest. They are related to (and slightly
weaker than) the notion of quasi-saturated maps and maps of Cartier type
studied by Tsuji.
