  This paper is concerned with the following logarithmic Schr\"{o}dinger
system: $$\left\{\begin{align} \ &\ -\Delta u_1+\omega_1u_1=\mu_1 u_1\log
u_1^2+\frac{2p}{p+q}|u_2|^{q}|u_1|^{p-2}u_1,\\ \ &\ -\Delta
u_2+\omega_2u_2=\mu_2 u_2\log u_2^2+\frac{2q}{p+q}|u_1|^{p}|u_2|^{q-2}u_2,\\ \
&\ \int_{\Omega}|u_i|^2\,dx=\rho_i,\ \ i=1,2,\\ \ &\ (u_1,u_2)\in
H_0^1(\Omega;\mathbb R^2),\end{align}\right.$$ where $\Omega=\mathbb{R}^N$ or
$\Omega\subset\mathbb R^N(N\geq3)$ is a bounded smooth domain,
$\omega_i\in\mathbb R$, $\mu_i,\ \rho_i>0,\ i=1,2.$ Moreover, $p,\ q\geq1,\
2\leq p+q\leqslant 2^*$, where $2^*:=\frac{2N}{N-2}$. By using a
Gagliardo-Nirenberg inequality and careful estimation of $u\log u^2$, firstly,
we will provide a unified proof of the existence of the normalized ground
states solution for all $2\leq p+q\leqslant 2^*$. Secondly, we consider the
stability of normalized ground states solutions. Finally, we analyze the
behavior of solutions for Sobolev-subcritical case and pass the limit as the
exponent $p+q$ approaches to $2^*$. Notably, the uncertainty of sign of $u\log
u^2$ in $(0,+\infty)$ is one of the difficulties of this paper, and also one of
the motivations we are interested in. In particular, we can establish the
existence of positive normalized ground states solutions for the
Br\'{e}zis-Nirenberg type problem with logarithmic perturbations (i.e.,
$p+q=2^*$). In addition, our study includes proving the existence of solutions
to the logarithmic type Br\'{e}zis-Nirenberg problem with and without the
$L^2$-mass $\int_{\Omega}|u_i|^2\,dx=\rho_i(i=1,2)$ constraint by two different
methods, respectively. Our results seems to be the first result of the
normalized solution of the coupled nonlinear Schr\"{o}dinger system with
logarithmic perturbation.
