Methods and apparatus are described for the use of a multi-dimensional logarithmic number system for hardware acceleration of inner product computations. These methods and apparatus may be used for any device that requires low-power, low-area and fast inner product computational units, such as, for example, deep neural network training and inference calculations on edge devices. In a particular embodiment, neural network training is performed using multi-dimensional logarithmic data representation, to obtain a set of neural network weight coefficients. Given the determined weight coefficients, the second base is optimized for multi-dimensional logarithmic data representation. This optimal representation may be used to perform inference by the neural network.