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Compared with the least-mean-square algorithm, the least mean pth power algorithm shows a better robustness performance against impulsive noises such as the α-stable noises. However, it still exhibits slow convergence rate and high kernel misadjustment. To overcome this drawback, a novel recursive logarithmic least mean pth (RLLMP) algorithm is proposed for the Volterra system identification under α-stable noise environments. Instead of minimizing the pth power, the new algorithm aims to minimize the pth logarithmic cost, which makes it more robust against impulsive interferences. Furthermore, to enhance tracking performance, an improved variable forgetting factor (IVFF) algorithm (IVFF-RLLMP) is proposed, which is based on the robust estimation of outliers. Simulation results are presented to demonstrate the improved performance of the RLLMP and IVFF-RLLMP.