  We provide a systematic treatment of the previously discovered power-enhanced
QED corrections to the leptonic decay $B_q\to \mu^+\mu^-$ ($q = d, s$) in the
framework of soft-collinear effective theory (SCET). Employing two-step
matching on SCET$_\text{I}$ and SCET$_\text{II}$, and the respective
renormalization group equations, we sum the leading-logarithmic QED corrections
and the mixed QED-QCD corrections to all orders in the couplings for the matrix
element of the semileptonic weak effective operator $\sim Q_9$. We propose a
treatment of the $B$-meson decay constant and light-cone distribution amplitude
in the presence of process-specific QED corrections. Finally we include
ultrasoft photon radiation and provide updated values of the non-radiative and
radiative branching fractions of $B_q \to \mu^+\mu^-$ decay that include the
double-logarithmic QED and QCD corrections.
