<p>In this paper, a non-convex logarithmic function is studied to compressed sensing from the perspective of theory, algorithm and computation. Firstly, we study the exact recovery properties of logarithmic function using the restricted isometry property (RIP) of measurement matrices, and the exact recovery conditions are derived. Secondly, we present an iterative thresholding algorithm to solve the regularization logarithmic minimization problem. A parameter-setting strategy is developed to set the proper parameters in proposed algorithm, and in this way, the algorithm will be adaptive and free from the selection of the proper parameters in each iteration. Finally, we provide some numerical simulations to show that the proposed algorithm outperforms the state-of-the-art algorithms in terms of the success rate and the number of measurements. Moreover, we also applied the proposed algorithm to the magnetic resonance imaging, and the simulation results verified the effectiveness of the proposed algorithm.</p>