This paper considers the parameter estimation problem under non-stationary environments in sensor networks. The unknown parameter vector is considered to be a time-varying sequence. To further promote estimation performance, this paper suggests a novel diffusion logarithm-correntropy algorithm for each node in the network. Such an algorithm can adopt both the logarithm operation and correntropy criterion to the estimation error. Moreover, if the error gets larger due to the non-stationary environments, the algorithm can respond immediately by taking relatively steeper steps. Thus, the proposed algorithm achieves smaller error in time. The tracking performance of the proposed logarithm-correntropy algorithm is analyzed. Finally, experiments verify the validity of the proposed algorithmic schemes, which are compared to other recent algorithms that have been proposed for parameter estimation.