Page 51 - The Master Handbook Of Acoustics

26 CHAPTER TWO

Table 2-1. Ways of expressing numbers (Continued).
Decimal Arithmetic Exponential
form form form
2
3
100,000 (100)(1,000) 10 + 10 = 10 2+3 = 10 5
4
2
100 10,000/100 10 /10 = 10 4–2 = 10 2
–1
4
5
10 100,000/10,000 10 /10 = 10 5–4 = 10 = 10
2
10 1 0 0 = 1 0 0 100 1/2 = 100 0.5
3
4.6416 1 0 0 100 1/2 = 100 0.333
4
31.6228 1 0 0 3 100 3/4 = 100 0.75
Logarithms
2
Representing 100 as 10 simply means that 10 × 10 = 100 and that 10 3
means 10 × 10 × 10 = 1,000. But how about 267? That is where loga-
2
rithms come in. It is agreed that 100 equals 10 . By definition you can
say that the logarithm of 100 to the base 10 = 2, commonly written log 10
100 = 2, or simply log 100 = 2, because common logarithms are to the
base 10. Now that number 267 needn’t scare us; it is simply expressed
as 10 to some other power between 2 and 3. The old fashioned way
was to go to a book of log tables, but with a simple hand-held calcula-
tor punch in 267, push the “log” button, and 2.4265 appears. Thus,
267 = 10 2.4265 , and log 267 = 2.4265. Logs are so handy because, as
Table 2-1 demonstrates, they reduce multiplication to addition, and
division to subtraction. This is exactly how the now-extinct slide rule
worked, by positioning engraved logarithmic scales.
Logs should be the friend of every audio worker because they are
the solid foundation of our levels in decibels. A level is a logarithm of
a ratio. A level in decibels is ten times the logarithm to the base 10
of the ratio of two power like quantities.

Decibels

A power level of a power W 1 can be expressed in terms of a reference
power W 2 as follows: