Recently, adaptive filtering algorithms have attracted much more attention in the field of signal processing.To make improvement in the performance of sigmoid function and geometric algebra based least mean square algorithm (GA-SVSLMS), this article proposed the novel approach to adaptive filtering with variable step size based on logarithmic function and geometric algebra (GA). First, the presented approach to adaptive filtering with variable step size based on geometric algebra represents the multi-dimensional signal as a GA multi-vector for the vectorization process. Second, by establishing a non-linear relationship between the step size and the error signal to achieve fast convergence and small steady error under reasonable computational complexity. Furthermore, we analyze the influence of the parameters in the step change factor on the proposed algorithm. Simulation conclusion reveals that the algorithm outperforms existing variable step size LMS algorithms based on other functions in terms of convergence rate and steady state error.