): An LNS can be considered as a floating-point number with the significand being always equal to 1 and a non-integer exponent.
On the other hand, the operations of addition and subtraction are more complicated and are calculated by the formulae where the "sum" function is defined by
The simplification of multiplication, division, roots, and powers is counterbalanced by the cost of evaluating these functions for addition and subtraction.
This added cost of evaluation may not be critical when using an LNS primarily for increasing the precision of floating-point math operations.
[1] Nicholas Kingsbury and Peter Rayner introduced "logarithmic arithmetic" for digital signal processing (DSP) in 1971.
[2] A similar LNS named "signed logarithmic number system" (SLNS) was described in 1975 by Earl Swartzlander and Aristides Alexopoulos; rather than use two's complement notation for the logarithms, they offset them (scale the numbers being represented) to avoid negative logs.
[4][1][5][6] The mathematical foundations for addition and subtraction in an LNS trace back to Zecchini Leonelli and Carl Friedrich Gauss in the early 1800s.
[7][8][9][10][11] In the late 1800s, the Spanish engineer Leonardo Torres Quevedo conceived a series of analogue calculating mechanical machines[12][13] and developed one that could solve algebraic equations with eight terms, finding the roots, including the complex ones.
One part of this machine called an "endless spindle" allowed the mechanical expression of the relation
A LNS has been used in the Gravity Pipe (GRAPE-5) special-purpose supercomputer[15] that won the Gordon Bell Prize in 1999.
A substantial effort to explore the applicability of LNSs as a viable alternative to floating point for general-purpose processing of single-precision real numbers is described in the context of the European Logarithmic Microprocessor (ELM).
[16][17] A fabricated prototype of the processor, which has a 32-bit cotransformation-based LNS arithmetic logic unit (ALU), demonstrated LNSs as a "more accurate alternative to floating-point", with improved speed.
Further improvement of the LNS design based on the ELM architecture has shown its capability to offer significantly higher speed and accuracy than floating-point as well.