Pipelined computation of very large word-length logarithmic number system (LNS) addition/subtraction requires a lot of hardware and long pipeline latency. This paper proposes a base-e exponential algorithm to simplify the exponential computation and to replace half of the pipeline stages by multiplication-and-accumulate operations. By using this approach, the circuit area and the pipeline latency of the previously proposed 64-bit basic LNS addition/subtraction unit can be reduced by 42.4% and 39.22%, respectively. Based on the base-e exponential algorithm approach, we also develop signed- digit (SD) exponential, SD discretization, and SD on-line logarithmic algorithms to further increase the throughput and to reduce the pipeline latency of the LNS computation. From our synthesis results, the throughput of the 64-bit LNS unit can be increased by 61.9% and the pipeline latency can be reduced by 55.0%, without increasing the hardware cost of the basic LNS unit. The circuit area of the 64-bit hardware-reduced LNS unit is estimated to be only 6.89 times the circuit area of a comparable 64-bit floating-point unit. We conclude that the proposed approaches have significantly improved the performance of the previously proposed LNS unit and have made a significant progress towards the implementation of very large word-length LNS arithmetic.