In this paper, we investigate the Fučík spectrum \(\Sigma _L\) associated with the logarithmic Laplacian. This spectrum is defined as the set of all pairs \((\alpha ,\beta ) \in \mathbb {R}^2\) for which the problem \(\begin{aligned} \left\{ \begin{aligned} L_\Delta u\,&= \alpha u^+-\beta u^- & ~~\text {in} ~~ \Omega , u&=0 & ~~\text {in} ~~\mathbb {R}^N\setminus \Omega , \end{aligned} \right. \end{aligned}\) admits a nontrivial solution u. Here, \(\Omega \subset \mathbb {R}^N\) is a bounded domain with \(C^{1,1}\) boundary, \(u^{\pm } = \max \{{\pm } u,0\}\) , and \(u = u^+ - u^-\) . We show that the lines \(\lambda _1^L \times \mathbb {R}\) and \(\mathbb {R} \times \lambda _1^L\) , where \(\lambda _1^L\) denotes the first eigenvalue of \(L_\Delta \) , lies in the spectrum \(\Sigma _L\) and are isolated within the spectrum. Furthermore, we establish the existence of the first nontrivial curve in \(\Sigma _L\) and analyze its qualitative properties, including Lipschitz continuity, strict monotonicity, and asymptotic behavior. In addition, we obtain a variational characterization of the second eigenvalue of the logarithmic Laplacian and show that all eigenfunctions corresponding to eigenvalues \(\lambda > \lambda _1^L\) are sign-changing. Finally, we address a nonresonance problem with respect to the Fučík spectrum \(\Sigma _L\) , employing variational methods and carefully overcoming the difficulties arising from the contrasting features of the first eigenvalue \(\lambda _1^L\) .
