  Using a combination of quantum Monte Carlo and exact methods, we study the
field-driven saturation transition of the two-dimensional $J$-$Q$ model, in
which the antiferromagnetic Heisenberg exchange $(J)$ coupling competes with an
additional four-spin interaction $(Q)$ that favors valence-bond solid order.
For small values of $Q$, the saturation transition is continuous, and is
expected to be governed by zero-scale-factor universality at its upper critical
dimension, with a specific form of logarithmic corrections to scaling (first
proposed by Sachdev \textit{et al.} [Phys. Rev. B \textbf{50}, 258 (1994)]).
Our results conform to this expectation, but the logarithmic corrections to
scaling do not match the form predicted by Sachdev \textit{et al.} We also show
that the saturation transition becomes first order above a critical coupling
ratio $(Q/J)_{\rm min}$ and is accompanied by magnetization
jumps---metamagnetism. We obtain an exact solution for $(Q/J)_{\rm min}$ using
a high magnetization expansion, and confirm the existence of the magnetization
jumps beyond this value of coupling using quantum Monte Carlo simulations.
