How do these both relate to one another? (Gain and dB's)

Decibel:

Gain:

Was that clear enough?

 

Jack is correct. But even simpler, dB is just a relative shorthand that is often used to describe gain. Nothing more. It is often used instead of the linear value because it leads to smaller numbers (instead of dealing with numbers like 1000 or 1 million you have 60 dB or 120 dB) and it makes certain mathematical operations simpler to do in your head.

Keep in mind that dB doesn't inherently have anything to do with gain, it is just a shorthand. People use dBs for other things with wide spreads in value for example isolation, noise, or relative level of harmonics in a signal.

 

taken a little further, there is ....
voltage gain = Voltage_Gain = 20 Log₁₀ x V_in /V_out in dB
and
Power gain = Power in Watts_Gain = 10 Log₁₀ x P_in /P_out in dB

note voltage is 20 x Log₁₀ and power is 10 x Log₁₀

for power gain and particularly when working with RF and amplifiers gain is usually related to a standard level ( a reference level)

this may be 0dBm = 1 mW or say 0dBW = 1W
so if using the 0dBm reference, a gain of 10dBm = 10mW, a gain of 20dBm = 100mW, a gain of 30dBm = 1000 mW (1W), a gain of 50dBm = 100,000 mW = 100W


you quickly see that this is a logarithmic scale

analogdesign's example of 1000 = 60dB is for a voltage remembering it is 20 x Log₁₀ and not 10 x Log₁₀ that is used for power


Dave

 

Sometimes it's tricky to get all the way down to the basics. Computers are just too nitpicky ...

Do you understand that "gain" implies a ratio ?
Alexander Graham Bell used the ratio 10 to 1 in his work and got a unit named after him, the "Bel" with just one "l" .
It stands for a power ratio of ten and is basic to sound measurement. We measure intensity of sound with logarithms because the ear is capable over such a wide range of loudness.

Try a search on that term, maybe it'll cement the idea in place for you.
https://www.bing.com/search?q=bel+u...id=B61CED28C1D647CE88938C06542557AD&FORM=QBRE