Chemotaxis is a phenomenon of the directed movement of cells in response to the concentration gradient of the chemical signal substance which is produced by cells. It can be divided into positive chemotaxis and negative chemotaxis according to the proximity or distance to the stimuli, which is meaningful to the survival and development of organisms. The study of chemotaxis is of great significance in the fields of biological oil recovery, ecological decontamination, biological population dynamics and the pathogenesis of infectious diseases. Therefore, a class of fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities and Logistic source is studied under homogeneous Neumann boundary conditions. The asymptotic behavior of the global bounded solution of the system is investigated by constructing an appropriate energy functional when it meets initial data and sensitivity functions satisfying some conditions. The result shows that the solution of the system will eventually converge to a constant steady state exponentially.