The logarithmic image processing (LIP) model is a mathematical framework which provides a specific set of algebraic and functional operations for the processing and analysis of intensity images valued in a bounded range. The LIP model has been proved to be physically justified by that it is consistent with the multiplicative transmittance and reflectance image formation models, and with some important laws and characteristics of human brightness perception. This article addresses the edge detection problem using the LIP-model based differentiation. First, the LIP model is introduced, in particular, for the gray tones and gray tone functions, which represent intensity values and intensity images, respectively. Then, an extension of these LIP model notions, respectively called gray tone vectors and gray tone vector functions, is studied. Third, the LIP-model based differential operators are presented, focusing on their distinctive properties for image processing. Emphasis is also placed on highlighting the main characteristics of the LIP-model based differentiation. Next, the LIP-Sobel based edge detection technique is studied and applied to edge detection, showing its robustness in locally small changes in scene illumination conditions and its performance in the presence of noise. Its theoretical and practical advantages over several well-known edge detection techniques, such as the techniques of Sobel, Canny, Johnson and Wallis, are shown through a general discussion and illustrated by simulation results on different real images. Finally, a discussion on the role of the LIP-model based differentiation in the current context of edge detection is presented.