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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 GL(2,Q). We show that rational subset membership for Baumslag-Solitar groups BS(1,q) with q> 2 is decidable and PSPACE-complete. To this end, we introduce a word representation of the elements of BS(1,q): their pointed expansion (PE), an annotatedarXiv:2006.11898v1 fatcat:y3q6p66235fwxbqumy6hrafb2q