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V_{1} = V_{in} and V_{2} = V_{out}. V_{2} > V_{1} or V_{out} > V_{in} means amplification. The dB value is positive. (+) V_{2} < V_{1} or V_{out} < V_{in} means damping. The dB value is negative. (−) V_{2} / V_{1} or V_{out} / V_{in} means the ratio. The amplification or the damping in dB is: L = 20 × log (voltage ratio V_{2} / V_{1}) in dB. V_{1} = V_{in} is the reference. 
In physics, attenuation is regarded as a positive value.
This naturally leads to sign errors when entering numbers.
3 dB ≡  1.414 times the voltage  (−)3 dB ≡  damping to the value 0.707 
6 dB ≡  2 times the voltage  (−)6 dB ≡  damping to the value 0.5 
10 dB ≡  3.162 times the voltage  (−)10 dB ≡  damping to the value 0.316 
12 dB ≡  4 times the voltage  (−)12 dB ≡  damping to the value 0.25 
20 dB ≡  10 times the voltage  (−)20 dB ≡  damping to the value 0.1 
Using voltage we get: Level in dB: L = 20 × log (voltage ratio) 
6 dB = twice the voltage 12 dB = four times the voltage 20 dB = ten times the voltage 40 dB = hundred times the voltage 
If we consider audio engineering, we are usually not interested in power.
Do not ask what power amplification means.
Leave that to the telephone companies or the transmitting aerials (antennas).
Power gain is really not used in audio engineering.
Do we really need power (energy) amplification?
Read the text at the bottom.
3 dB ≡  2 times the power  (−3) dB ≡  damping to the value 0.5 
6 dB ≡  4 times the power  (−6) dB ≡  damping to the value 0.25 
10 dB ≡  10 times the power  (−10) dB ≡  damping to the value 0.1 
12 dB ≡  16 times the power  (−12) dB ≡  damping to the value 0.0625 
20 dB ≡  100 times the power  (−20) dB ≡  damping to the value 0.01 
Using power we get: Level in dB: L = 10 × log (power ratio)
3 dB = twice the power 6 dB = four times the power 10 dB = ten times the power 20 dB = hundred times the power 
If you search for the amplification ratio, given the dB value,
then go to the program dB calculation
Amplification (Gain) and Damping (Loss)
To use the calculator, simply enter a value. The calculator works in both directions of the ↔ sign. 
In audio technique the following "power or energy amplification " is rather unusual.
Voltage/Pressure amplification ratio 
1 
1.414 = √2 
2 
3.16 = √10 
4 
10 
20 
40 
100 
1000 
Increasing of x dB  0  3  6  10  12  20  26  32  40  60 
Power/Intensity amplification ratio 
1 
1.414 = √2 
2 
3.16 = √10 
4 
10 
20 
40 
100 
1000 
Increasing of y dB  0  1.5  3  5  6  10  13  16  20  30 

To use the calculator, simply enter a value. The calculator works in both directions of the ↔ sign. 
The voltage is always given as RMS value  but that is not valid for electric power.
There is also the reference power P_{0} = 1 milliwatt or 0.001 watt ≡ 0 dB_{m}
Level in psycho acoustics as subjectivly perceived loudness (volume)
The vague human feeling of the double loudness (volume) is specified
with about 6 to 10 dB. This personal feeling is not an exactly measurable value.
Conversion Factor, Ratio, or Gain to a Level Value (Decibels dB)
Amplifier conversion − Convert decibels to voltage gain / loss
Calculator Voltage Gain − Voltage Loss and Power Gain − Power Loss
Voltage gain in dB 
Power gain in dB 
Voltage ratio = amplification factor (voltage) 
Power ratio = amplification factor (power) 
V_{1} = V_{in} and V_{2} = V_{out}. V_{2} > V_{1} or V_{out} > V_{in} means amplification. The dB value is positive. (+) V_{2} < V_{1} or V_{out} < V_{in} means damping. The dB value is negative. (−) V_{2} / V_{1} or V_{out} / V_{in} means the ratio. The amplification or the damping in dB is: L = 20 × log (voltage ratio V_{2} / V_{1}) in dB. V_{1} = V_{in} is the reference. 

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