Here, we study the dynamics of the singularly perturbed logistic difference equation with two different continuous arguments. First of all, local stability of the fixed points is investigated by analyzing the corresponding characteristic equations of the linearized equations. Secondly, we illustrate that the considered system exhibits Hopf bifurcation. A discretized analogue of the original system is obtained using the method of steps. Local stability and bifurcation analysis of the discretized system are investigated. Explicit conditions for the occurrence of a variety of complex dynamics such as fold and Neimark-Sacker bifurcations are reached. We compare the results with those of the associated difference equation with continuous argument when the perturbation parameter $\epsilon \longrightarrow 0$ and with those of the logistic delay differential equation with two different delays when $\epsilon \longrightarrow 1$. Finally, numerical simulations including Lyapunov exponent, bifurcation diagrams and phase portraits are carried out to confirm the theoretical analysis obtained and to illustrate more complex dynamics of the system.