  Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We
define new conjecturally integer-valued counts of $\mathbb{A}^1$-curves in
$(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten
invariants of maximal tangency along $E$ via a formula analogous to the
multiple cover formula for disk counts. A conjectural relationship to genus 0
local BPS numbers is described and verified for del Pezzo surfaces and curve
classes of arithmetic genus up to 2. We state a number of conjectures and
provide computational evidence.
