In this paper, we study the following coupled nonlinear logarithmic Hartree
system \begin{align*} \left\{ \displaystyle \begin{array}{ll} \displaystyle
-\Delta u+ \lambda_1 u =\mu_1\left( -\frac{1}{2\pi}\ln(|x|) \ast u^2
\right)u+\beta \left( -\frac{1}{2\pi}\ln(|x|) \ast v^2 \right)u, & x \in ~
\mathbb R^2, \vspace{.4cm}\\ -\Delta v+ \lambda_2 v =\mu_2\left(
-\frac{1}{2\pi}\ln(|x|) \ast v^2 \right)v +\beta\left( -\frac{1}{2\pi}\ln(|x|)
\ast u^2 \right)v, & x \in ~ \mathbb R^2, \end{array} \right.\hspace{1cm}
\end{align*} where β,μi​,λi​ (i=1,2) are positive constants,
∗ denotes the convolution in R2. By considering the constraint
minimum problem on the Nehari manifold, we prove the existence of ground state
solutions for β>0 large enough. Moreover, we also show that every
positive solution is radially symmetric and decays exponentially