What’s the relationship between gain (dB) and power (watts)?

Resistance is the term used to denote the opposition to the flow of current in a circuit or material. It depends on the resistivity and dimensions of the material. The resistance offered by a material to the flow of direct current and alternating current remains the same. It is denoted by the letter R.

Impedance

Impedance is the opposition to alternating current. It is a combination of resistance and reactance. It is represented by a complex number and has a real and an imaginary part. The real part represents resistance and the imaginary part represents the reactance. Reactance can be capacitive, reactive or a combination of both.

source: https://www.electricalclassroom.com/resistance-vs-impedance/

@rixthetrick - thank you sir!  Love your posts and will refer to them again as needed.

Amplification can be somewhat described as if you had a small electrical signal controlling a valve on a larger electrical signal, as the smaller signal swings from positive to negative, in higher and lower amplitude from zero, it opens and closes the valve proportionally, positive and negative. 

This is where my technical ignorance seems significant.  Maybe I’d need a textbook to understand this:


Well, think about how much an amp can do.  Amps can't have infinite power, or output voltage, right?  They are limited devices.  Every one of them. What limits them?  Among many things, the voltage available to the output circuits. That is the absolute limit of output.

Your gain is 20x (28 dB) for instance.  OK, but your maximum voltage out is 10 Volts peak to peak.

1 volt pk-pk in x 20 = 20 Volts pk-pk. 

With this amp, 20 volts out is not possible.   You need a bigger amp.  What you'll get instead is a clipped signal that goes up to 10 and hangs there until the input voltage goes back down.

@erik_squires - Thank you. That’s helpful.

Well, think about how much an amp can do. Amps can't have infinite power, or output voltage, right? They are limited devices. Every one of them. What limits them? Among many things, the voltage available to the output circuits. That is the absolute limit of output.

Your gain is 20x (28 dB) for instance. OK, but your maximum voltage out is 10 Volts peak to peak.

1 volt pk-pk in x 20 = 20 Volts pk-pk.

With this amp, 20 volts out is not possible.   You need a bigger amp. What you'll get instead is a clipped signal that goes up to 10 and hangs there until the input voltage goes back down.

Look at it this way.

A Designer has design requirements.

Design an amp that produces 250 watts per channel with an input of x mV into an 8 ohm load that will produce the 250 watts per channel max without clipping.  an input impedance of y Ohms, output impedance of P Ohms, frequency range of 20 Hz to 20 kHz, etc.

This gives you the gain of the amp.  As MillerCarbon and many other have told you, it then is a straight forward calculation.

There are typical standards in many amp designs.   They call for an input voltage of x MV in order to produce max (before clipping) output of xxx watts per channel into an 8 ohm load.  So,  most audio amps have a set gain with the input sensitivity voltage of x mV to produce that max output.

This is electronics engineering 101.

The amp specs will tell you the amps gain based on the designed for input voltage to get that max power output.

The amp in question, has a gain switch that changes the amps gain.  They are either adding another gain stage in the amp via the switch to allow for lower input sensitivity or they are adjusting the input sensitivity and adding or removing a gain stage or just adjusting the input sensitivity.

either way, what MillerCarbon and some others have described is pretty accurate.

Its not rocket science, but it does require some knowledge of electronic design.

If you start with the required input voltage sensitivity, the required output power rating, the required load, input impedance requirement, output impedance requirements, the amp will have a calculated gain.

If you start with a set gain requirement and power output, into a set load, you still need to know what the input sensitivity is.

Pre-amps are pretty similar, except that there are standard pre-amp input voltage sensitivities, and max output voltage requirements (so you don't blow up your amp and speakers). 

anyway, I hope this didn't confuse you further.

enjoy