  Let $f$ be the germ of a real analytic function at the origin in
$\mathbb{R}^n $ for $n \geq 2$, and suppose the codimension of the zero set of
$f$ at $\mathbf{0}$ is at least $2$. We show that $\log |f|$ is
$W^{1,1}_{\operatorname{loc}}$ near $\mathbf{0}$. In particular, this implies
the differential inequality $|\nabla f |\leq V |f|$ holds with $V \in
L^1_{\operatorname{loc}}$. As an application, we derive an inequality relating
the {\L}ojasiewicz exponent and singularity exponent for such functions.
