  It has been known for many years that soft radiation can give potentially
large double-logarithm corrections to $p_\perp$ broadening of a high-energy
particle traveling through QCD matter, but that this soft radiation correction
can be absorbed into an effective value $\hat q_{\rm eff}$ for the medium
$p_\perp$-broadening parameter $\hat q$. Here "soft" means high energy compared
to medium scales but soft compared to the original high-energy particle
traveling through the medium. A similar situation arises in the case of soft
corrections to hard splitting of a high-energy particle, such as hard
$g{\to}gg$, where double logarithms can also be absorbed using the same
effective $\hat q_{\rm eff}$. In this paper, I study whether the same holds
true for potentially large, subleading, *single*-logarthim corrections. The
correspondence is more indirect for single logarithms, but I show (in the
large-$N_{\rm c}$ limit) that single logarithms from soft radiation in the case
of $p_\perp$ broadening also determine the single logarithms from soft
radiative corrections to hard $g{\to}gg$ splitting. Along the way, there is an
interesting variation of the original BDMPS-Z calculation of splitting rates in
the $\hat q$ approximation. I also discuss how, for soft-radiative corrections
to hard splitting processes, there are two different types of "$\hat q_{\rm
eff}$" that come into play, which differ by "$i\pi$" terms that multiply single
logarithms.
