  In this paper, we study the radial symmetry and monotonicity of nonnegative
solutions to nonlinear equations involving the logarithmic
Schr$\ddot{\text{o}}$dinger operator $(\mathcal{I}-\Delta)^{\log}$
corresponding to the logarithmic symbol $\log(1 + |\xi|^2)$, which is a
singular integral operator given by $$(\mathcal{I}-\Delta)^{\log}u(x)
=c_{N}P.V.\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N}}\kappa(|x-y|)dy,$$
where $c_{N}=\pi^{-\frac{N}{2}}$,
$\kappa(r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}\mathcal{K}_{\frac{N}{2}}(r)$ and
$\mathcal{K}_{\nu}$ is the modified Bessel function of second kind with index
$\nu$. The proof hinges on a direct method of moving planes for the logarithmic
Schr$\ddot{\text{o}}$dinger operator.
