  We construct the $\mathbb{A}^1$-local stable motivic homotopy categories of
fs log schemes. For schemes with the trivial log structure, our construction is
equivalent to the original construction of Morel-Voevodsky. We prove the
localization property. As a consequence, we obtain the Grothendieck six
functors formalism for strict morphisms of fs log schemes. We extend
$\mathbb{A}^1$-invariant cohomology theories of schemes to fs log schemes. In
particular, we define motivic cohomology, homotopy $K$-theory, and algebraic
cobordism of fs log schemes. For any fs log scheme log smooth over a scheme, we
express cohomology of its boundary in terms of cohomology of schemes.
