Let \(\mathbb {X}\) be a commutative Banach algebra with identity. This work investigates the existence and stability of equilibrium solutions for the nonlinear reaction-diffusion equation: \(\begin{aligned} u_{t}=u_{xx}+wu+\textrm{Ln}(u^2)u, \end{aligned}\) where u is a function from \(\mathbb {R}\times \mathbb {R}^+\) to \(\mathbb {X}\) , \(w\in \mathbb {X}\) is a fixed parameter, and \(\textrm{Ln}\) denotes the logarithmic function defined on a specific bounded open set of \(\mathbb {X}\) . We derive an explicit formula, using exponential functions, for a family of equilibrium solutions that decay to zero at infinity. The instability of these equilibria is established through two methods. Firstly, we perform a detailed spectral stability analysis by examining the spectrum of the linearized operator around the equilibrium solutions. Secondly, we analyze the behavior of particular curves of solutions that are perturbations of the equilibrium solutions. This approach is also used to demonstrate the nonlinear instability of any non-trivial equilibrium solution of the equation. Finally, we show that the nonlinear instability implies the linear instability by utilizing a specific Lie symmetry of the equation.