Level (logarithmic quantity)

"Level quantity" redirects here. For other uses, see Level measurement.

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 of a quantity Q, denoted LQ, is defined by


Level of a field quantity

Level of a field quantity, denoted LF, is defined by


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 LF, is therefore


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 LP, is defined by


For the level of a power quantity, the base of the logarithm is r = e2.[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

If F is a root-power quantity:

If P is a power quantity:

If the power quantity P is equal to F2, and if the reference value of the power quantity, P0, is equal to F02, the levels LF and LP 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.


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

See also


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