  The natural maximal and minimal functions commute pointwise with the
logarithm on $A_\infty$. We use this observation to characterize the spaces
$A_1$ and $RH_\infty$ on metric measure spaces with a doubling measure. As the
limiting cases of Muckenhoupt $A_p$ and reverse H\"older classes, respectively,
their behavior is remarkably symmetric. On general metric measure spaces, an
additional geometric assumption is needed in order to pass between $A_p$ and
reverse H\"older descriptions. Finally, we apply the characterization to give
simple proofs of several known properties of $A_1$ and $RH_\infty$, including a
refined Jones factorization theorem. In addition, we show a boundedness result
for the natural maximal function.
