Bases of primitive linear groups

Let V be a finite vector space and G GL(V) a linear group. A base of G is a set of vectors whose pointwise stabiliser in G is trivial. We prove that if G is irreducible and primitive on V, then G has a base of size at most 18logG/logV+c, where c is an absolute constant. This verifies part of a conjecture of Pyber on base sizes of primitive permutation groups.