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This paper addresses the power control problems of Cognitive Radio (CR) under transmission power and interference temperature constraints. First, we propose the interference constraint which ensures that the Quality of Service (QoS) standards for primary users is considered and a non-cooperative game power control model. Based on the proposed model, we developed a logical utility function based on the Signal-to-Interference-Noise Ratio (SINR) and a novel algorithm that is suitable for CR network power control. Then, the existence and uniqueness of the Nash Equilibrium (NE) in our utility function are proved by the principle of game theory and the corresponding optimizations. Compared to traditional algorithms, the proposed one could converge to an NE in 3-5 iterative operations by setting an appropriate pricing factor. Finally, simulation results verified the stability and superiority of the novel algorithm in flat-fading channel environments.