  Let G be a permutation group, acting on a set \Omega of size n. A subset B of
\Omega is a base for G if the pointwise stabilizer G_(B) is trivial. Let b(G)
be the minimal size of a base for G. A subgroup G of Sym(n) is large base if
there exist integers m and r \geq 1 such that Alt(m)^r \unlhd G \leq Sym(m) \wr
Sym(r), where the action of Sym(m) is on k-element subsets of {1,...,m} and the
wreath product acts with product action. In this paper we prove that if G is
primitive and not large base, then either G is the Mathieu group M24 in its
natural action on 24 points, or b(G) \leq \lceil \log n\rceil+1. Furthermore,
we show that there are infinitely many primitive groups G that are not large
base for which b(G) > log n + 1, so our bound is optimal.
