Deep neural networks virtually dominate the domain of most modern vision systems, providing high performance at a cost of increased computational complexity. Since for those systems it is often required to operate both in real-time and with minimal energy consumption (e.g., for wearable devices or autonomous vehicles, edge Internet of Things (IoT), sensor networks), various network optimisation techniques are used, e.g., quantisation, pruning, or dedicated lightweight architectures. Due to the logarithmic distribution of weights in neural network layers, a method providing high performance with significant reduction in computational precision (for 4-bit weights and less) is the Power-of-Two (PoT) quantisation (and therefore also with a logarithmic distribution). This method introduces additional possibilities of replacing the typical for neural networks Multiply and ACcumulate (MAC – performing, e.g., convolution operations) units, with more energy-efficient Bitshift and ACcumulate (BAC). In this paper, we show that a hardware neural network accelerator with PoT weights implemented on the Zynq UltraScale + MPSoC ZCU104 SoC FPGA can be at least 1.4x more energy efficient than the uniform quantisation version. To further reduce the actual power requirement by omitting part of the computation for zero weights, we also propose a new pruning method adapted to logarithmic quantisation.