This paper introduces Logarithmic Wavelets. The concept is born from the application of the Logarithmic Image Processing (LIP) and Symetric Logarithmic Image Processing (S-LIP) models to wavelets. It leads to a coupled logarithmic-multiscale image processing approach. After a review of classical aspects of wavelet theory, theoretical aspects of the Logarithmic Wavelet and Logarithmic Wavelet Transform are presented. It is demonstrated that the chosen LIP and S-LIP Models constitute adapted frameworks to achieve such developments. The most classical applications of wavelets, like compression, contour detection, de-noising are studied in this novel logarithmic context, which produces most of the time better results than the linear approach.