Unsolvability of certain transcendental equations and the corresponding braid monodromy

We prove that the equations [Formula: see text] and [Formula: see text] are not solvable in elementary functions. The proof is based on the methods of topological Galois theory. The core of our argument is the analysis of the braid group representation arising from the monodromy action on the roots of the equation. We show that the image of this representation, the monodromy group, is unsolvable, which implies the unsolvability of the equation itself in elementary functions.