Representing DNNs with low-precision numbers is a promising approach that enables the efficient acceleration of large-scale deep neural networks (DNNs). However, previous methods typically keep a copy of weights in high precision for weight updates during training. Directly training over low-precision weights still remains an unsolved problem because of the complex interactions between low-precision number systems and the underlying learning algorithms. To address this problem, we develop a low-precision training framework, termed LNS-Madam, in which we jointly design a logarithmic number system (LNS) and a multiplicative weight update training method (Madam). LNS-Madam yields low quantization error during weight update, leading to a stable convergence even if the precision is limited. By replacing SGD or Adam with the Madam optimizer, training under LNS requires less weight precision during the updates while preserving the state-of-the-art prediction accuracy.