  Given any compact, Hausdorff space $K$ and $1<p<\infty$, we compute the
Szlenk and $w^*$-dentability indices of the spaces $C(K)$ and $L_p(C(K))$. We
show that if $K$ is compact, Hausdorff, scattered, $CB(K)$ is the
Cantor-Bendixson index of $K$, and $\xi$ is the minimum ordinal such that
$CB(K)\leqslant \omega^\xi$, then $Sz(C(K))=\omega^\xi$ and
$Dz(C(K))=Sz(L_p(C(K)))= \omega^{1+\xi}.$
