  We consider the rational subset membership problem for Baumslag-Solitar
groups. These groups form a prominent class in the area of algorithmic group
theory, and they were recently identified as an obstacle for understanding the
rational subsets of $\text{GL}(2,\mathbb{Q})$.
  We show that rational subset membership for Baumslag-Solitar groups
$\text{BS}(1,q)$ with $q\ge 2$ is decidable and PSPACE-complete. To this end,
we introduce a word representation of the elements of $\text{BS}(1,q)$: their
pointed expansion (PE), an annotated $q$-ary expansion. Seeing subsets of
$\text{BS}(1,q)$ as word languages, this leads to a natural notion of
PE-regular subsets of $\text{BS}(1, q)$: these are the subsets of
$\text{BS}(1,q)$ whose sets of PE are regular languages. Our proof shows that
every rational subset of $\text{BS}(1,q)$ is PE-regular.
  Since the class of PE-regular subsets of $\text{BS}(1,q)$ is well-equipped
with closure properties, we obtain further applications of these results. Our
results imply that (i) emptiness of Boolean combinations of rational subsets is
decidable, (ii) membership to each fixed rational subset of $\text{BS}(1,q)$ is
decidable in logarithmic space, and (iii) it is decidable whether a given
rational subset is recognizable. In particular, it is decidable whether a given
finitely generated subgroup of $\text{BS}(1,q)$ has finite index.
