  Let $(X,\omega)$ be a compact K\"ahler manifold. We prove the existence and
uniqueness of solutions to complex Monge-Amp\`ere equations with prescribed
singularity type. Compared to previous work, the assumption of small unbounded
locus is dropped, and we work with general model type singularities. We state
and prove our theorems in the context of big cohomology classes, however our
results are new in the K\"ahler case as well. As an application we confirm a
conjecture by Boucksom-Eyssidieux-Guedj-Zeriahi concerning log-concavity of the
volume of closed positive $(1,1)$-currents. Finally, we show that log-concavity
of the volume in complex geometry corresponds to the Brunn-Minkowski inequality
in convex geometry, pointing out a dictionary between our relative
pluripotential theory and $P$-relative convex geometry. Applications related to
stability and existence of csck metrics are treated elsewhere.
