Last updated: 2026-07-01

Decibel ⇄ Ratio

The decibel (dB) is a logarithmic unit that expresses the ratio between two values of a physical quantity. It is not an absolute unit: a decibel figure is always relative to a reference, and the same ratio produces a different decibel value depending on whether the quantity is a power quantity or an amplitude (field) quantity. This converter moves between a linear ratio and its decibel value in both directions, under either law.

Definition

For a power quantity — acoustic intensity, radiated power, optical power — the level in decibels of a power $P$ relative to a reference $P_0$ is:

For an amplitude or field quantity — voltage, sound pressure, electric-field strength, current — the level uses a factor of 20 instead of 10:

Here $\log_{10}$ is the base-10 logarithm. The two laws are consistent with each other: because power is proportional to the square of amplitude ($P \propto V^2$), and $\log(V^2) = 2\log(V)$, an amplitude ratio contributes exactly twice the decibels of the corresponding power ratio. Reporting the same physical change as a power ratio or as an amplitude ratio gives the same decibel figure.

Inverting the formula

To recover the linear ratio from a decibel value, invert the logarithm. For a power ratio:

and for an amplitude ratio:

A ratio is always positive, so converting from a ratio to decibels requires $r > 0$. A ratio of exactly 1 gives 0 dB under either law, and a ratio below 1 gives a negative decibel value (attenuation).

Common values

The two rules of thumb worth memorising are that +3 dB roughly doubles power and +6 dB roughly doubles amplitude. Each is exact only in the limit; $10^{3/10} \approx 1.995$, not 2.

Worked example

Suppose an amplifier is specified as having a voltage gain of 40 dB, and you want the linear voltage ratio. Voltage is an amplitude quantity, so the 20·log law applies and its inverse is $r = 10^{L/20}$:

So 40 dB of voltage gain means the output amplitude is 100 times the input. If the same 40 dB figure were describing a power gain, the inverse would use the 10·log law, $r = 10^{40/10} = 10^{4} = 10\,000$ — a reminder that the quantity type must be known before the number can be interpreted.

Related conversions

To convert absolute power figures between watts, horsepower, and other units, use the Power Unit Converter. For frequency-based ratios, see the Frequency Converter.