  We study the Lane-Emden system involving the logarithmic Laplacian: $$
\begin{cases} \ \mathcal{L}_{\Delta}u(x)=v^{p}(x) ,& x\in\mathbb{R}^{n},\\ \
\mathcal{L}_{\Delta}v(x)=u^{q}(x) ,& x\in\mathbb{R}^{n}, \end{cases} $$ where
$p,q>1$ and $\mathcal{L}_{\Delta}$ denotes the Logarithmic Laplacian arising as
a formal derivative $\partial_s|_{s=0}(-\Delta)^s$ of fractional Laplacians at
$s=0.$ By using a direct method of moving planes for the logarithmic Laplacian,
we obtain the symmetry and monotonicity of the positive solutions to the
Lane-Emden system. We also establish some key ingredients needed in order to
apply the method of moving planes such as the maximum principle for
anti-symmetric functions, the narrow region principle, and decay at infinity.
Further, we discuss such results for a generalized system of the Lane-Emden
type involving the logarithmic Laplacian.
