# Level (logarithmic quantity)

In the International System of Quantities, the **level** of a quantity is the logarithm of the ratio of the value of that quantity to a reference value of the same quantity.^{[1]}^{[2]} Examples are the various types of sound level: sound power level (literally, the level of the sound power, abbreviated SWL), sound exposure level (SEL), sound pressure level (SPL) and particle velocity level (SVL).^{[3]}

## Mathematical definitions

### Level

Level of a quantity *Q*, denoted *L*_{Q}, is defined by

where

*r*is the base of the logarithm;*Q*is the root-power quantity;*Q*_{0}is the reference value of*Q*.

### Level of a field quantity

Level of a *field* quantity, denoted *L*_{F}, is defined by

where

*F*is the field quantity;*F*_{0}is the reference value of*F*.

For the level of a field quantity, the base of the logarithm is *r* = e.

### Level of a root-power quantity

A root-power quantity is a field quantity. The Level of a *root-power* quantity, denoted *L*_{F}, is therefore

where

*F*is the root-power quantity;*F*_{0}is the reference value of*F*.

For the level of a root-power quantity, the base of the logarithm is *r* = e.

### Level of a power quantity

Level of a *power* quantity, denoted *L*_{P}, is defined by

where

*P*is the power quantity;*P*_{0}is the reference value of*P*.

For the level of a power quantity, the base of the logarithm is *r* = e^{2}.^{[4]}

## Units of level

### Power level

The neper, bel, and decibel (one tenth of a bel) are units of level that are often applied to such quantities as power, intensity, or gain.^{[5]} The neper, bel, and decibel are defined by

- Np = 1;
- B = 1/2 ln 10 Np;
- dB = 0.1 B = 1/20 ln 10 Np.

If *F* is a root-power quantity:

If *P* is a power quantity:

If the power quantity *P* is equal to *F*^{2}, and if the reference value of the power quantity, *P*_{0}, is equal to *F*_{0}^{2}, the levels *L*_{F} and *L*_{P} are equal.

### Frequency level

The octave is a unit of level (specifically "frequency level", for *r* = 2) though that concept is seldom seen outside of the standard.^{[6]} A semitone is one twelfth of an octave.

## Standardization

The level and its units are defined in ISO 80000-3.

## See also

## References

- ↑ ISO 80000-3:2006,
*Quantities and units*, Part 2: Space and Time - ↑ W. M. Carey, Sound Sources and Levels in the Ocean, IEEE J Oceanic Eng 31:61-75(2006)
- ↑ ISO 80000-8:2007,
*Quantities and units*, Part 8: Acoustics - ↑ Ainslie, M. A. A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound. Acoustics Today, 23 February 2015
- ↑ Barry Taylor (1995).
*Guide for the Use of the International System of Units (SI): The Metric System*. Diane Publishing Co. p. 28. - ↑ ANSI/ASA S1.1-2013,
*Acoustical Terminology*