  We prove a formula expressing the $K$-theoretic log Gromov-Witten invariants
of a product of log smooth varieties $V \times W$ in terms of the invariants of
$V$ and $W$. The proof requires introducing log virtual fundamental classes in
$K$-theory and verifying their various functorial properties. We introduce a
log version of $K$-theory and prove the formula there as well.
