This paper considers the problem of reconstructing sparse or compressible signals from one-bit quantized measurements. We study a new method that uses a log-sum penalty function, also referred to as the Gaussian entropy, for sparse signal recovery. Also, in the proposed method, sigmoid functions are introduced to quantify the consistency between the acquired one-bit quantized data and the reconstructed measurements. A fast iterative algorithm is developed by iteratively minimizing a convex surrogate function that bounds the original objective function, which leads to an iterative reweighted process that alternates between estimating the sparse signal and refining the weights of the surrogate function. Connections between the proposed algorithm and other existing methods are discussed. Numerical results are provided to illustrate the effectiveness of the proposed algorithm.