We consider the logarithmic Fekete problem, which consists of placing a fixed number of points on the unit sphere in \(\mathbb {R}^d\), in such a way that the product of all pairs of mutual Euclidean distances is maximized or, equivalently, so that their logarithmic energy is minimized. Using tools from Computational Algebraic Geometry, we find and classify all critical configurations for this problem when considering at most six points in every dimension d. In particular, our approach gives new proofs of several key results appearing in the literature, with the benefit of using a unified approach.