  Let $X$ be a nonsingular variety defined over an algebraically closed field
of characteristic $0$, and $D$ be a free divisor. We study the motivic Chern
class of $D$ in the Grothendieck group of coherent sheaves $G_0(X)$, and
another class defined by the sheaf of logarithmic differentials along $D$. We
give explicit calculations of the difference of these two classes when: $D$ is
a divisor on a nonsingular surface; $D$ is a hyperplane arrangement whose
affine cone is free.
