<p>This paper proposes a novel multiplicative logistic map derived from the rule of multiplicative calculus and introduces an additional parametric freedom that fundamentally extends its dynamical capabilities. The theoretical and numerical analysis confirm that this map undergoes a period-doubling bifurcation cascade into chaos as rigorously validated by stability analysis, bifurcation diagrams, transversality conditions and stability conditions. Crucially, compared to the classical logistic map, it exhibits a significantly broader chaotic region and an expanded output range beyond [0,1]. Cobweb and time-series plots visually confirm these enhanced and complex behaviors. Moreover, owing to its greater parametric flexibility and wider chaotic dynamics, the multiplicative logistic map is a highly suitable candidate for advanced encryption applications. Experimental results and comparative analysis demonstrate that the image encryption algorithm based on the proposed map exhibits strong resistance to statistical attacks and superior parameter robustness in practical implementations.</p>