NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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A logarithm 
is fundamentally an exponent
applied
to a specific base 
. 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
is
). The inverse of a logarithm is called
anantilogarithm or
antilog.
For any positive number 
, we
have
for any valid base 
. This is just an identity arising from the definition
of the logarithm, but it is sometimes useful in manipulating
formulas.
When the base is not specified, it is normally assumed to be
, i.e.,
. This is the common logarithm.
Base 2 and base 
logarithms have their own special notation:
(The use of 
for base 
logarithms is common in computer
science. In mathematics, it may denote a base logarithm.) By far the most
common bases are , , and
. Logs
base are
called natural
logarithms. They are
''natural'' in the sense that
while the derivatives of logarithms to other bases are not quite so
simple:
(Prove this as an exercise.) The inverse of the natural
logarithm
is
of course the exponential
function 
,
and 
is its
own derivative.
In general, a logarithm has an
integer part and a fractional part. The integer part is called the
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 [6].
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 = 0.0043
Logarithms were used in the days before computers to
performmultiplication 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 tradeoff between memory (for the log
tables) and computation. Nowadays, large numbers are multiplied
using FFT fast-convolution
techniques.
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