## Logarithms

A*logarithm*is fundamentally an

*exponent*applied to a specific

*base*to yield the argument . That is, . The term ``logarithm'' can be abbreviated as ``log''. The base is chosen to be a positive real number, and we normally only take logs of positive real numbers (although it is ok to say that the log of 0 is ). The inverse of a logarithm is called an

*antilogarithm*or

*antilog*; thus, is the antilog of in the base .

For any positive number , we have

*i.e.*, . This is the

*common logarithm*. Base 2 and base logarithms have their own special notation:

*natural logarithm*s. They are ``natural'' in the sense that

*characteristic*of the logarithm, and the fractional part is called the

*mantissa*. These terms were suggested by Henry Briggs in 1624. ``Mantissa'' is a Latin word meaning ``addition'' or ``make weight''--something added to make up the weight [28]. The following Matlab code illustrates splitting a natural logarithm into its characteristic and mantissa:

>> x = log(3) x = 1.0986 >> characteristic = floor(x) characteristic = 1 >> mantissa = x - characteristic mantissa = 0.0986 >> % Now do a negative-log example >> x = log(0.05) x = -2.9957 >> characteristic = floor(x) characteristic = -3 >> mantissa = x - characteristic mantissa = 0.0043Logarithms were used in the days before computers to perform

*multiplication of large numbers*. Since , one can look up the logs of and in tables of logarithms, add them together (which is easier than multiplying), and look up the antilog of the result to obtain the product . Log tables are still used in modern computing environments to replace expensive multiplies with less-expensive table lookups and additions. This is a classic trade-off between memory (for the log tables) and computation. Nowadays, large numbers are multiplied using FFT fast-convolution techniques.

### Changing the Base

By definition, . Taking the log base of both sides gives### Logarithms of Negative and Imaginary Numbers

By*Euler's identity*, , so that

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