The decibel (dB) is most generally associated with sound levels, especially for those of us in acoustics. However, acoustics has no sole claim to the use of this term. For example, there is no reason why the decibel could not be used to express values in frequency, weight, height, distance, the national debt, or to any measurement that has two values of like units.

This idea may seem strange to many, but if one understands the decibel, the above is completely logical. In actuality, the decibel is defined mathematically to express a *ratio*. And because the decibel is defined mathematically it need not be limited to expressing ranges in the intensities of sounds only. For example, in the financial world, the difference between $1,000 and $10,000,000 can be expressed as 40 decibels re: $1.00 (assuming that money is power, and the power formula is used).

#### In acoustics the decibel is defined from the logarithm

Because the intensity of the loudest tolerable sound to the intensity of the faintest audible sound is approximately 100,000,000,000,000 (a hundred trillion) to 1, mathematicians have devised a technique to reduce writer’s cramp by reducing the size of the numbers involved. The technique used is called the *logarithm*, and it is from this that the decibel has been defined. More specifically, *the dB is the log difference ratio of one value*

*to another*. In other words, find the ratio between the two values and convert that into a logarithm. As can be imagined, the decibel is most useful when there are large differences between the values measured and when expressed in large numbers.

#### From mathematics

The logarithm (abbreviated “log”) is an exponent. It tells how many times a number must be multiplied by itself. For example 10^{3} equals 1000 (10 x 10 x 10 = 1000). In this case the number 10 is the base and the 3 is the exponent. The base is usually indicated by a subscript to the word log. *(In this example 10 after Log should be a subscript but the editing program for this blog does not provide a mechanism for dropping it down to show it as a subscript).*

###### Log_{10}1000 = 3

In exponential form this would be written as,

10^{3} = 1000

#### Linear vs. logarithmic scales

A *linear *(interval) scale consists of successive units that are obtained by adding a given unit to each successive number. For example,

Unit of 1: 1, 2, 3, 4, 5, 6, 7………………………………. n

Unit of 2: 2, 4, 6, 8, 10, 12, 14…………………………… n

A *logarithmic* (ratio) scale is generated by successive multiplying of some unit.

Unit of 1: 1, 1, 1, 1, 1, 1, 1, 1……………………………….. 1

Unit of 2: 1, 2, 4, 8, 16, 32, 64, ………………………… n

Unit of 10: 1, 10, 100, 1000, 10000, 100000……. n

Using the unit of 10 as an example, the logarithmic scale can also be written as follows:

10^{0} = 1

10^{1} = 10

10^{2} = 100

10^{3} = 1,000

——–

——–

10^{7} = 10,000,000

10^{8} = 100,000,000

etc.

When the logarithmic scale is written with the exponents it is referred to an exponential scale, as noted above in “From Mathematics.” When written this way note that the exponents form an interval series. This makes it possible to change a ratio (logarithmic) series into an interval (linear) series, and thus reduce the size of the numbers involved.

10^{9} rather than 1,000,000,000

Logarithm to the base 10 is also referred to as the common logarithm and is used in the formulae for sound energy. Consequently, the base is often omitted from logarithmic formulae related to sound, but understood to be 10. (For frequency, 2 would be the base, rather than 10).

#### Bel

Suppose that a sound intensity ratio of 10,000:1 exists (one sound is 10,000 times greater than the other). In this example the logarithm is defined as the number of Bels representing that ratio. The number of Bels for this ratio is 4; a ratio of 100:1 is 2 Bels, etc. The formula for Bel is:

B = log R

Where B is the number of Bels and R is the associated intensity ratio between the two sounds.

If it were assumed that the smallest ratio is 1:1, the number of Bels difference between the two sounds would be 0. While the Bel provides a means for comparing sound intensities, it seems too large a unit for practical use. Using the Bel suggests a range of only 14 comparisons when measuring sound intensity differences over the range of human response. Because of this, the decibel (dB) was suggested. As the name implies, the decibel is one-tenth of a Bel. Consequently, there are ten times as many decibels as Bels in a given intensity ratio. This extends the range of measurement comparisons to 140 in the sound intensity range. The corresponding formula for decibel is:

dB = 10 log R

The comparison between Bels and decibels is as follows:

Intensity Ratio Bel Decibel Equivalent

1:1 0 0

10:1 1 10

100:1 2 20

1000:1 3 30

——– — —–

10,000,000,000,000:1 13 130

**Non-hearing example;**

As an example of the use of the decibel for something other than sound levels, the difference between the weight of a young person of 100 lbs and a medium-sized elephant of 10,000 lbs is 20 dB re 1 lb (assuming that weight is equated to power – perhaps a stretch somewhat, but nevertheless helpful to demonstrate alternate uses of the decibel).

*Calculation*: (the power formula is used because weight is considered power in this example)

dB = 10 log R

dB = 10 log (10,000 lb/1b)

dB = 10 log 10^{4} (log 10^{4} = 4.0000)

dB = 10 x 4.0000

dB = 40 re 1 lb

dB = 10 log (100 lb/1 lb)

dB = 10 log 10^{2} (log 10^{2} = 2.0000)

dB = 10 x 2.0000

dB = 20 re 1 lb

The dB difference between the two weights is 40 dB – 20 dB = 20 dB re 1 lb.