  We show that the minimal base size $b(G)$ of a finite primitive permutation
group $G$ of degree $n$ is at most $2 (\log |G|/\log n) + 24$. This bound is
asymptotically best possible since there exists a sequence of primitive
permutation groups $G$ of degrees $n$ such that $b(G) = \lfloor 2 (\log
|G|/\log n) \rceil - 2$ and $b(G)$ is unbounded. As a corollary we show that a
primitive permutation group of degree $n$ that does not contain the alternating
group $\mathrm{Alt}(n)$ has a base of size at most $\max\{\sqrt{n} , \ 25\}$.
