AbstractWe study the existence and uniqueness of positive solutions of the nonlinearfractional relaxation differential equation where Dα/1 is the Caputo-Hadamard fractional derivative of order 0 < α ≤ 1. In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct an appropriate mapping and employ the Schauder fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. Finally, an example is given to illustrate our results.