This paper deals with the following Schrödinger system with logarithmic coupling terms: \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_1 + \lambda _1 u_1 = \mu _1 u_1\log u_1^2 + \beta u_1\log (u_1^2+u_2^2), & x\in \mathbb {R}^N, \\ -\Delta u_2 + \lambda _2 u_2 = \mu _2 u_2\log u_2^2 + \beta u_2\log (u_1^2+u_2^2), & x\in \mathbb {R}^N, \end{array} \right. \end{aligned}\) where \(N \ge 1\) , \(\lambda _1,\lambda _2,\mu _1,\mu _2,\beta \in \mathbb {R}\) are constants. Based on the variation idea for weakly lower semi-continuous functionals in [40], we prove that the problem has a positive ground state solution if \(\mu _1,\mu _2<0\) and \(\beta >\max \{-\mu _1,-\mu _2\}\) . Besides, we develop a new version of general minimax principle restricting on two dimensional paths for lower semi-continuous functionals, and then apply it to find a positive solution with higher energy if \(\mu _1,\mu _2>0\) and \(\max \{-\mu _1,-\mu _2\}<\beta <\beta _{\lambda _1,\lambda _2,\mu _1,\mu _2}\) for some positive constant \(\beta _{\lambda _1,\lambda _2,\mu _1,\mu _2}\) depending on \(\lambda _1,\lambda _2,\mu _1\) and \(\mu _2\) .
