<p>This paper investigates the multiplicity of solutions for a fourth-order two-point boundary value problem involving a sign-changing logarithmic nonlinearity. The indefinite nature of the nonlinearity prevents the direct use of classical variational methods. To overcome this difficulty, we employ a refined logarithmic Sobolev inequality combined with the Nehari manifold technique. Under suitable assumptions on the coefficient function, we prove the existence of at least two distinct nontrivial solutions. Additionally, we provide a numerical simulation based on a finite difference scheme and a fixed-point iteration, which confirms the theoretical findings and illustrates the profiles of the two solutions. The results highlight the effectiveness of variational methods in handling non-standard nonlinearities and demonstrate the agreement between analytical and numerical approaches.</p>