  Logarithmic number systems (LNS) are used to represent real numbers in many
applications using a constant base raised to a fixed-point exponent making its
distribution exponential. This greatly simplifies hardware multiply, divide and
square root. LNS with base-2 is most common, but in this paper we show that for
low-precision LNS the choice of base has a significant impact.
  We make four main contributions. First, LNS is not closed under addition and
subtraction, so the result is approximate. We show that choosing a suitable
base can manipulate the distribution to reduce the average error. Second, we
show that low-precision LNS addition and subtraction can be implemented
efficiently in logic rather than commonly used ROM lookup tables, the
complexity of which can be reduced by an appropriate choice of base. A similar
effect is shown where the result of arithmetic has greater precision than the
input. Third, where input data from external sources is not expected to be in
LNS, we can reduce the conversion error by selecting a LNS base to match the
expected distribution of the input. Thus, there is no one base which gives the
global optimum, and base selection is a trade-off between different factors.
Fourth, we show that circuits realized in LNS require lower area and power
consumption for short word lengths.
