This paper is concerned with the existence and properties of normalized solutions to the following logarithmic Schrödinger system \(\begin{aligned} \left\{ \begin{aligned}&-\Delta u+\lambda _1 u=\mu _1u\log u^2+\theta \alpha |u|^{\alpha -2}|v|^{\beta }u&\text {in }{{\mathbb {R}}^N},\\&-\Delta v+\lambda _2 v=\mu _2v\log v^2+\theta \beta |v|^{\beta -2}|u|^{\alpha }v&\text {in }{{\mathbb {R}}^N}, \\&\ \int _{{{\mathbb {R}}^N}} \left|u\right|^2 ~\textrm{d} x=a^2 \quad \text { and }\quad \int _{ {{\mathbb {R}}^N}} \left|v\right|^2 ~\textrm{d} x=b^2, \end{aligned} \right. \end{aligned}\) where \(N\ge 2\) , \(a,b>0\) , \(\mu _1,\mu _2, \theta \in {{\mathbb {R}}}\setminus \left\{ 0\right\} \) , and the exponents \(\alpha>1,\beta >1\) satisfy the Sobolev critical and subcritical conditions \(\begin{aligned} \begin{aligned} 2<\alpha +\beta< \infty , \quad \text {when } N=2,\\ 2 <\alpha +\beta \le 2^*,\quad \text {when } N\ge 3. \end{aligned} \end{aligned}\) The parameters \(\lambda _1, \lambda _2\in {{\mathbb {R}}}\) will arise as Lagrange multipliers that are not prior given. For the focusing case \( \mu _1,\mu _2>0\) , we establish various results concerning the existence, multiplicity, and stability/instability of normalized solutions when \(\theta >0\) . Furthermore, we demonstrate that as \(\theta \rightarrow 0^+\) , these normalized solutions of the logarithmic Schrödinger system converge to constant multiples of a specific solution with physical significance, known as the Gausson, which is the unique positive ground state solution of the logarithmic scalar field equation \(\begin{aligned} -\Delta u = u\log u^2~~~\text {in }{{\mathbb {R}}^N}, \ \ u \in H^1({{\mathbb {R}}}^N). \end{aligned}\) Besides, we also give a criteria for global existence and finite time blow-up in the associated dispersive system. Finally, for the defocusing case \( \mu _1,\mu _2<0\) , we prove a nonexistence result when \(\theta <0\) and find a radially symmetric sign-changing normalized solution when \(\theta >0\) . This paper combines several approaches and gives a rather complete picture of the logarithmic Schrödinger system.
