Adding to the answer by @Robert Israel (which I'm sure he knows, but didn't mention):
Because log is strictly monotonically increasing, this problem is equivalent to $maximize \left( \|x\|_\infty \right)$ subject to the same constraints.
That is a concave programming problem, i.e., the minimization of a concave function ($ -\|x\|_\infty$) subject to convex constraints. If the constraints are compact, there is a global optimum at an extreme of the constraints, although there may be additional global optima not at an extreme. There may also be one or more local optima which are not global optima.
Presuming an upper bound can be placed on $x$, then by introduction of binary variables, this can be converted to a Mixed-Integer Linear Programming (MILP) problem, and solved to global optimality by any off the shelf MILP solver, such as Gurobi or CPLEX, among many others. It is NP hard, so it might take a while (and require some memory), but the business model of the companies which develop and sell MILP solvers is predicated on the ability of their customers to routinely solve NP hard problems using their solvers.