TABLE OF CONTENTS A number system based on logarithms can simplify multiplication, division, roots, and powers. When logarithms are used, multiplication and division are reduced to addition and subtraction, respectively, and powers and roots are reduced to multiplication and division, respectively. On the other hand, add and subtract operations become more complex. Another major problem is deriving logarithms and antilogarithms quickly and accurately enough to allow conversions to and from the conventional number representations. These conversions always involve approximations, resulting in inaccuracies. Therefore, binary logarithms can be useful only in arithmetic units dedicated to special applications where very few conversions are required but many multiplications and divisions are executed; e.g., real-time digital filters.
Let a number A be represented by a sign digit S A and a logarithm E A that includes an integer part and a fractional part
requiring a total of n=1+ k+l bits. The sign S A is set to 0 if A is positive, and to 1 if A is negative. E A is the logarithm of the absolute value of A; i.e., E A ?log 2 A. The interpretation rule for S A E A is thus
The base of the exponent may, in general, be different from 2.
To represent numbers smaller than 1, negative logarithms are needed. For this purpose, we may use the two s complement representation or a biased representation, which takes the general form
Commonly...
