Summary: This paper is concerned with the following logarithmic Schrödinger system \[ \begin{cases} - \Delta u_1 + \omega_1 u_1 = \mu_1 u_1 \log u^2_1 + \cfrac{2p}{p+q} |u_2|^q |u_1|^{p-2} u_1, \\ - \Delta u_2 + \omega_2 u_2 = \mu_2 u_2 \log u^2_2 + \cfrac{2q}{p+q} |u_1|^p |u_2|^{q-2} u_2, \\ \displaystyle\int_\Omega |u_i|^2 dx = \rho_i, \quad i = 1, 2, \\ (u_1, u_2) \in H^1_0 (\Omega; \mathbb{R}^2), \end{cases} \] where \(\Omega = \mathbb{R}^N\) or \(\Omega \subset \mathbb{R}^N\) \((N \geqslant 3)\) is a bounded smooth domain, and \(\omega_i \in \mathbb{R}\), \(\mu_i, \rho_i > 0\) for \(i = 1, 2\). Moreover, \(p, q \geqslant 1\), and \(2 \leqslant p + q \leqslant 2^*\), where \(2^* := \frac{2N}{N - 2}\). By using a Gagliardo-Nirenberg inequality and a careful estimation of \(u \log u^2\), firstly, we provide a unified proof of the existence of the normalized ground state solution for all \(2 \leqslant p + q \leqslant 2^*\). Secondly, we consider the stability of normalized ground state solutions. Finally, we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent \(p + q\) approaches \(2^*\). Notably, the uncertainty of the 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 state solutions for the Brézis-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 Brézis-Nirenberg problem with and without the \(L^2\)-mass constraint \(\int_\Omega |u_i|^2 dx = \rho_i\) \((i = 1, 2)\) by two different methods, respectively. Our results seem to be the first result of the normalized solution of the coupled nonlinear Schrödinger system with logarithmic perturbations.