Logarithm is a great invention; it is an inverse operation of power (continued product).

Logarithm is denoted as:

Y = log ^a_x

Where: ‘a’ is the base(底数), ‘x’ is an antilogarithm(真数), and ‘Y’ is the logarithm.

Bel or dB (decibel) is a logarithm form of ratio. It is a pure value without any physical units. It is defined as:

Bel = Log ¹⁰ _x

Also simplified denoted as:

Bel = lg _x

Where: ‘x’ is the linear form of ratio.

This form will be used in the following texts.

For Bel is a very big value for most of the applications, tenth of Bel is defined as decibel (dB), denoted as:

dB = 10 lg _x

Some commonly used dB values and its according linear values (most values are the approximation values):

0 dB = 1

1 dB = 1.26

2 dB = 1.5(exactly 1.58 but conventionally approximately as 1.5)

3 dB = 2.0(exactly 1.995 but conventionally approximately as 2.0)

6 dB = 4.0(exactly 3.98 but conventionally approximately as 4.0)

10 dB = 10.0

20 dB = 100.0

-1 dB = 0.79

-2 dB =0.63

-3 dB = 0.5

-10 dB = 0.1

-20 dB = 0.01

Gain is often used in electronics. Gain is amplification multiples, but, specially, stand for that of power, rather than voltage or current.

Power Gain A(p)(dB) = 10 lg_(Po/Pi)

For voltage of current the value is the double of that of power. The reason is that power is proportional to the power of voltage or current for the same resistance.

P = I ₂ R = V ₂/R

Voltage Gain A(V)(dB) = 20 log ¹⁰ _{( Vo / Vi)} = 20 lg _{( Vo / Vi)}

Current Gain A(I)(dB) = 20 log ¹⁰ _{(Io / Ii)} = 20 lg _{( Io / Ii )}

Most common used units derived from dB:

Signal strength dBm is defined as the reference of

0 dBm = 1 mW

Here the unit is not a pure value any more, it stand for a power.

Aerials gain dBi is defined as the reference of standard all directions aerials:

0 dBi = the power of a standard all directions aerials

Sound pressure level:

0 dBSPL = 0.00002 N/m²

That is the sound human could just hear.

130 dBSPL is the terrible sound that is harmful to human auditory sense.

Some applications:

-3 dB bandwidth:

Cut-off frequency is a frequency the power decreased to half of the maximum of the signal power as the frequency increased. And for the voltage is 0.707 of the Maximum voltage.

CMRR (Common-Mode Rejection Ratio):

dB =  10 lg_{(Differential-Mode gain / Common-Mode gain)}

Signal Noise/Distortion Ratio:

dB = 10 lg_{(Signal Power/(Noise + Distortion Power))}

PSRR (Power Supply Rejection Ratio):

dB =  10 lg_{(Signal Gain /Power Supply Ripple gain)}

Dynamic Range:

dB =  10 lg_{(Max Signal Power/ Min Signal Power)}