  We propose a delayed Mittag-Leffler type matrix function with logarithm,
which is an extension of the classical Mittag-Leffler type matrix function with
logarithm and delayed Mittag-Leffler type matrix function. With the help of the
delayed Mittag-Leffler type matrix function with logarithm, we give an explicit
form a of solutions to nonhomogeneous Hadamard type fractional time-delay
linear differential equations. Moreover, we study existence uniqueness and
stability in Ulam-Hyers sense of the Hadamard type fractional time-delay
nonlinear equations.
