  In this paper we show that if $G$ is a primitive subgroup of $S_{n}$ that is
not large base, then any irredundant base for $G$ has size at most $5 \log n$.
This is the first logarithmic bound on the size of an irredundant base for such
groups, and is best possible up to a small constant. As a corollary, the
relational complexity of $G$ is at most $5 \log n+1$, and the maximal size of a
minimal base and the height are both at most $5 \log n.$ Furthermore, we deduce
that a base for $G$ of size at most $5 \log n$ can be computed in polynomial
time.
