# 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 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 LQ, 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 LF, 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 LF, 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 LP, 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.[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. 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 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.[5] A semitone is one twelfth of an octave.

## Standardization

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