Considering the non-linear properties of the human visual system, many non-linear operators and models have been developed, particularly the logarithmic image processing (LIP) model proposed by Jourlin and Pinoli, which has been proved to be physically justified in several laws of the human visual system and has been successfully applied in image processing areas. Recently, several modifications based on this logarithmic mathematical framework have been presented, such as parameterized logarithmic image processing (PLIP), pseudo-logarithmic image processing, homomorphic logarithmic image processing. In this paper, a new single parameter logarithmic model for image processing with an adaptive parameter-based Sobel edge detection algorithm is presented. On the basis of analyzing the distributive law, the subtractive law, and the isomorphic property of the PLIP model, the five parameters in PLIP are replaced by a single parameter to ensure the completeness of the model and physical constancy with the nature of an image, and then an adaptive parameter-based Sobel edge detection algorithm is proposed. By using an image noise estimation method to evaluate the noise level of image, the adaptive parameter in the single parameter LIP model is calculated based on the noise level and grayscale value of a corresponding image area, followed by the single-parameter LIP-based Sobel operation to overcome the noise-sensitive problem of classical LIP-based Sobel edge detection methods, especially in the dark area of an image, while retaining edge sensitivity. Compared with the classical LIP and PLIP model, the given single parameter LIP achieves satisfactory results in noise suppression and edge accuracy.