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
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:
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 [
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 = 0.0043
Logarithms 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
which tells how to convert the base from

to

, that is, how to
convert the log base

of

to the log base

of

. (Just
multiply by the log base

of

.)
By
Euler's identity,

, so that
from which it follows that for any

,

.
Similarly,

, so that
and for any imaginary number

,

,
where

is real.
Finally, from the polar representation

for
complex numbers,
where

and

are real. Thus, the log of the magnitude of
a complex number behaves like the log of any positive
real number,
while the log of its phase term

extracts its phase
(times

).
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