Image processing technologies such as image enhancement generally utilize linear arithmetic operations to manipulate images. Recently, Jourlin and Pinoli successfully used the logarithmic image processing (LIP) model for several applications of image processing such as image enhancement and segmentation. In this paper, we introduce a parameterized LIP (PLIP) model that spans both the linear arithmetic and LIP operations and all scenarios in between within a single unified model. We also introduce both frequency- and spatial-domain PLIP-based image enhancement methods, including the PLIP Lee's algorithm, PLIP bihistogram equalization, and the PLIP alpha rooting. Computer simulations and comparisons demonstrate that the new PLIP model allows the user to obtain improved enhancement performance by changing only the PLIP parameters, to yield better image fusion results by utilizing the PLIP addition or image multiplication, to represent a larger span of cases than the LIP and linear arithmetic cases by changing parameters, and to utilize and illustrate the logarithmic exponential operation for image fusion and enhancement.