This paper investigates the stability properties of Caputo-Hadamard fractional differential equations. We first analyze the asymptotic behavior and rigorously prove a specific convergence rate for these equations. Then, a novel stability criterion called logarithmic Mittag-Leffler stability is proposed. By employing the fixed-point theorem in an innovative Banach space equipped with a designed weighted norm, we demonstrate that when the linearization spectrum of the Caputo-Hadamard fractional differential equation lies within a prescribed sector, the equilibrium point of the equation exhibits logarithmic Mittag-Leffler stability. This result leads to a version of Lyapunov's first method for Caputo-Hadamard fractional differential equations, demonstrating its stability in the presence of logarithmic memory.