124
D. Rocchesso: Sound Processing
exponential
logarithm
decibel
rms level
Neper number
The function f (x) = a
x
is called exponential with base a.
Given these preliminary definitions and properties, we define the logarithm
of y with base a
x = log
a
y ,
(16)
as the inverse function of y = a
x
. In other words, it is the exponent that must
be given to the base in order to get the argument y. Since the power a
x
has been
defined only for a > 0 and it gives always a positive number, the logarithm is
defined only for positive values of the independent variable y.
Logarithms are very useful because they translate products and divisions
into sums and differences, and power operations into multiplications. Simply
stated, by means of the logarithms it is possible to reduce the complexity of
certain operations. In fact, the properties E1­3 allow to write down the following
properties:
L1 : log
a
xy = log
a
x + log
a
y
L2 : log
a
x
y
= log
a
x - log
a
y
L3 : log
a
x
y
= y log
a
x .
In sound processing, the most interesting logarithm bases are 10 and 2. The
base 10 is used to define the decibel (symbol dB) as a ratio of two quantities. If
the quantities x and y are proportional to sound pressures (e.g., rms level), we
say that x is wdB larger than y if x > y > 0 and
w = 20 log
10
x
y
.
(17)
When the quantities x and y are proportional to a physical power (or intensity),
their ratio in decibel is measured by using a factor 10 instead of 20
8
in (17).
The base 2 is used in all branches of computer sciences, since most com-
puting systems are based upon binary representations of numbers (see the ap-
pendix A.9). For instance, the number of bits that is needed to form an address
in a memory of 1024 locations is
log
2
1024 = 10 .
(18)
In Octave/Matlab, the logarithms of x having base 2 and 10 are indicated
with log2(x) and log10(x), respectively. Fig. 6 shows the curves of the loga-
rithms in base 2 and 10. From these curves we can intuitively infer how, in any
base, log 1 = 0, and how the function approaches - (minus infinity) as the
argument approaches zero.
Given a logarithm expressed in base a, it is easy to convert it in the logarithm
expressed in another base b. The formula that can be used is
log
b
x =
log
a
x
log
a
b
.
(19)
A base of capital importance in calculus is the Neper number e, a transcen-
dental number approximately equal to 2.7183. As we will see in appendix A.7.1,
8
In acoustics [86], the power is proportional to the square of a pressure. Therefore, applying
property L3, we fall back into the definition (17).
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