In the previous work [24], de Queiroz and Shahgholian investigated the regularity of solutions to the obstacle problem with a singular logarithmic forcing term \(\begin{aligned} -\Delta u = \log u \, \chi _{\{u>0\}} \quad \text {in} \quad \Omega , \end{aligned}\) where \(\chi _{\{u>0\}}\) denotes the characteristic function of the set \(\{u>0\}\) and \(\Omega \subset \mathbb {R}^n\) ( \(n \ge 2\) ) is a smooth bounded domain. The solution is obtained by solving the minimization problem for the functional \(\begin{aligned} \mathscr {J}(u):=\int _{\Omega }\left( \frac{|\nabla u|^2}{2}-u^+ (\log u-1)\right) \, dx, \end{aligned}\) where \(u^+=\max \{0,u\}\) . In this paper, based on the regularity of the solution, we establish the \(C^{1,\alpha }\) regularity of the free boundary \(\Omega \cap \partial \{u>0\}\) near regular points for some \(\alpha \in (0,1)\) . The logarithmic forcing term becomes singular near the free boundary \(\Omega \cap \partial \{u>0\}\) and lacks scaling properties, which are crucial for studying the regularity of the free boundary. Despite these challenges, we draw overall inspiration from the epiperimetric inequality method introduced by Weiss [32]. Central to our approach is the introduction of a new type of energy contraction. This allows us to obtain energy decay, which in turn ensures the uniqueness of blow-up limit and subsequently leads to the regularity of the free boundary.
