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. 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).

Mathematical definitions

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

Level

Level of a quantity Q, denoted L_Q, is defined by

$L_{Q}=\log _{r}\!\left({\frac {Q}{Q_{0}}}\right)\!,$

where

• r is the base of the logarithm;
• Q is the root-power quantity;
• Q₀ is the reference value of Q.

Level of a field quantity

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

$L_{F}=\ln \!\left({\frac {F}{F_{0}}}\right)\!,$

where

• F is the field quantity;
• F₀ 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

$L_{F}=\ln \!\left({\frac {F}{F_{0}}}\right)\!,$

where

• F is the root-power quantity;
• F₀ 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

$L_{P}=\log _{e^{2}}\!\left({\frac {P}{P_{0}}}\right)={\frac {1}{2}}\ln \!\left({\frac {P}{P_{0}}}\right)\!,$

where

• P is the power quantity;
• P₀ is the reference value of P.

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

Units of level

Power level

See also: Decibel § Conversions, and Neper § Units

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. 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:

$L_{F}=\ln \!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {Np} =2\log _{10}\!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {B} =20\log _{10}\!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {dB} .$

If P is a power quantity:

$L_{P}={\frac {1}{2}}\ln \!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {Np} =\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {B} =10\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {dB} .$

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 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. A semitone is one twelfth of an octave.

See also

• Decibel § Definition
• Field, power, and root-power quantities
• Logarithmic scale
• Sound level (disambiguation)