  The article is devoted to the study of mappings that satisfy the so-called
inverse Poletsky inequality. We consider mappings of quasiextremal distance
domains, domains with a locally quasiconformal boundary, and domains which are
regular in the sense of prime ends onto domains with a locally quasiconformal
boundary, regular domains, or domains which are locally H\"{o}lder equivalent
to a half-ball on its boundary. For such mappings, we have obtained the
H\"{o}lder logarithmic continuity in some neighborhood of its boundary points.
