Arithmetic and Logic in Computer Systems

10

Operations through Logarithms

10.1 MULTIPLICATION AND ADDITION IN LOGARITHMIC SYSTEMS

Given an X value, from a read only memory (ROM) one can read the value of log₂X where X is used as the address and log₂X is the data located at that address. Also, given log₂X one can find the antilogarithm by reading from ROM the X value. Here log₂X is used as the address line and X is the data at that address. Note that two separate ROMs are needed for the logarithm and antilogarithm functions, while in the pencil and paper case one table can be shared.

In a logarithmic system multiplication can be completed through addition. First consider unsigned numbers only. Because log₂(X × Y) = log₂X + log₂Y, to calculate X × Y, we first find log₂X and log₂Y. After adding them to get a sum, we find the antilogarithm of the sum. That is knowing log₂(X × Y), we look for the product of X × Y. Also, in the logarithmic system division can be carried on through subtraction. Because log₂(X/Y) = log₂X − log₂Y, to calculate X/Y we first find log₂X and log₂Y. After subtracting log₂Y from log₂X we find the antilogarithm of the difference.

Take into consideration the signed numbers. If P = X × Y, P can be represented by where S_P is the sign of product P, and 2^log₂|P| = |P|. Given log₂|P|, 2^log₂|P| denotes the antilogarithm function to find |P| from log₂|P|. Note that sign

S_P = S_X ⊕ S_Y,

and

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