2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226348We provide a refined approach to the classical Magnus and Fer expansion, unveiling a new structure by using the language of dendriform and pre-Lie algebras. The recursive formula for the logarithm of the solutions of the equations X=1+ta<X and Y=1-tY> a in A[[t]] is provided, where (A,<,>) is a dendriform algebra. Then, we present the solutions to these equations as an infinite product expansion of exponentials. Both formulae involve the pre-Lie product naturally associated with the dendriform structure. Several applications are presented.13 pages, LaTeX. Terminology issues fixed. Page setup problems fixed on pdf file. Final version as to appear in Foundations of Computational MathematicsCombinatoricsMathematical PhysicsClassical Analysis and ODEsRings and Algebras05E99; 16W99text