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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
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).