  We investigate the quantum phase transition in an $S = 1/2$ dimerized
Heisenberg antiferromagnet in three spatial dimensions. By performing
large-scale quantum Monte Carlo simulations and detailed finite-size scaling
analyses, we obtain high-precision results for the quantum critical properties
at the transition from the magnetically disordered dimer-singlet phase to the
antiferromagnetically ordered N\'eel phase. This transition breaks O($N$)
symmetry with $N = 3$ in $D = 3 + 1$ dimensions. This is the upper critical
dimension, where multiplicative logarithmic corrections to the leading
mean-field critical properties are expected; we extract these corrections,
establishing their precise forms for both the zero-temperature staggered
magnetization, $m_s$, and the N\'eel temperature, $T_N$. We present a scaling
Ansatz for $T_N$, including logarithmic corrections, which agrees with our data
and indicates exact linearity with $m_s$, implying a complete decoupling of
quantum and thermal fluctuation effects even arbitrarily close to the quantum
critical point. We also demonstrate the predicted $N$-independent leading and
subleading logarithmic corrections in the size-dependence of the staggered
magnetic susceptibility. These logarithmic scaling forms have not previously
been identified or verified by unbiased numerical methods and we discuss their
relevance to experimental studies of dimerized quantum antiferromagnets such as
TlCuCl$_3$.
