Bel

The bel is a logarithmical unit used to express relative difference in intensity or power between two signals, usually acoustic. Because of the fact that the Bel unit is too large for everyday use most of the times it is used as the more convenient decibel (abbreviated to dB).

Decibel is equal to ten times the common logarithm of the ratio of the two levels.

${di\!f\!\!f}=\cdot \log _{10}\left({\frac {P_{2}}{P_{1}}}\right)$

In case of the decibel we have a 10 factor:

${di\!f\!\!f}=10\cdot \log _{10}\left({\frac {P_{2}}{P_{1}}}\right)$

There is a second definition of the decibel. In acoustics, it is used to express the ratio in sound pressure, because power is roughly proportional to the square of effective sound pressure.

Let 'p' be pressure, 'P' power.

${di\!f\!\!f}=\cdot \log _{10}\left({\frac {P_{2}}{P_{1}}}\right)$

${di\!f\!\!f}=\cdot \log _{10}\left({\frac {p_{2}^{2}}{p_{1}^{2}}}\right)$

${di\!f\!\!f}=2\cdot \log _{10}\left({\frac {p_{2}}{p_{1}}}\right)$ because $\log _{10}(a^{2})=2\cdot \log _{10}(a)$

In this scale, it means an increase of 20 dB is equal to a ten-fold ratio, or 6 dB a two-fold ratio ($20\cdot \log _{10}(2)$ is approximately equal to 6).