In this paper, we studied the dynamic behavior of a Cournot-type duopoly model with homogeneous goods, a convex log-linear price function, and quadratic cost functions. By using the Lambert $W$ function, we derived explicit expressions for the fixed points and showed that the model has exactly three fixed points. We conducted a local stability analysis to identify the stability regions for each fixed point. Furthermore, we demonstrated that the system undergoes a sequence of period-doubling bifurcations, which give rise to stable cycles of period 2 and period 4. We showed that the period-4 cycle loses its stability through a Neimark-Sacker bifurcation, where an attracting invariant closed curve emerges. We also derived the topological normal forms associated with the period-doubling bifurcation points. To confirm the presence of chaotic behavior, we computed the largest Lyapunov exponents. Finally, numerical simulations, along with bifurcation analyses performed using MatContM, validated our theoretical results. These findings extend and enhance previous studies in the literature, particularly the results in ^[1,2].