Summary: In this paper, we study the following logarithmic Schrödinger-Bopp-Podolsky system \[ \left\{\begin{array}{ll} -\varepsilon^2\Delta u+V(x)u-\phi u=u \log u^2,& \text{ in }\mathbb{R}^3,\\ -\varepsilon^2\Delta \phi +\varepsilon^4\Delta^2\phi =4\pi u^2, &\text{ in }\mathbb{R}^3, \end{array}\right. \] where \(\varepsilon\) is a small positive parameter and \(V(x)\in C(\mathbb{R}^3,\mathbb{R})\). Under the global condition on potential \(V(x)\), we prove the existence of positive solution \(u_{\varepsilon}\in H^1(\mathbb{R}^3)\) of above system for \(\varepsilon >0\) small enough by applying the Variational Methods developed by Szulkin for the functional which is a sum of a \(C^1\) functional with a convex lower semicontinuous functional. Moreover, we also investigate the concentration behavior of \(\{u_{\varepsilon}\}\) as \(\varepsilon \rightarrow 0\).