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

IN the end MC’s point is one all should take to heart. Doubling the volume -- all things equal - requires 10X the power.

Don’t mean to be difficult but no, that’s not quite right. Sound volume is how loud we call something we perceive or hear. That’s why I resist measuring and object to people saying they listen at so and so many dB. Instead I prefer to say I listen at a satisfying level, or if I want to get across that it is good and loud I like to say Supertramp Loud.

It sounds goofy but at least it is clear I know the difference between volume level and SPL. Sound pressure level is what we measure with decibels. Literally the pressure difference between the compression and rarefactions of the sound wave.

The difference is that we can say a 10dB increase requires ten times the power. Because both decibels and watts are quantifiable. Volume however is totally unquantifiable. Is 10 dB twice as loud? According to who? Some can barely hear a 2dB change. For others that is a big jump and they want a volume control with finer increments. For others anything past a certain point is simply "too loud".

But your main point is definitely one people will do well to learn. To play even just 3dB louder requires twice the power. 3dB is not that much of a difference. But it requires twice the amp to do it. This more than anything else is why people will do themselves such a big favor to ignore low sensitivity speakers. You can get them. They can be made to work. But the math is stacked against you.

If the amp is rated 300Watt at 8Ω, gain set to 23dB, the amplifier need 3.5Volt input signal in order to output 300Watt at 8Ω,
If the gain set to 29dB, the amplifier need 1.75Volt input to output 300Watt at 8Ω.
Its depends on the max. output voltage of your preamp, set the gain on your power amp accordingly.

@millercarbon - thank you.  I like this:

That is the difference between power and gain. Gain is a multiplier. Power is a ceiling. You cannot get any more power from your amp no matter what gain you set it on.

@timlub - thank you. This is helpful and simple:

With higher gain, it takes less of a turn of the volume knob before you hit the amplifiers full output.
With the lower gain setting, you need to twist it a bit further.

@erik_squires - now this seems relevant to me:

That is, put in 0.1V peak to peak input signal and you should get 2V peak to peak on the output. 

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails. So, if your rails are +- 20V, that's your peak output. With 20x gain (around 28 dB) this means your maximum input voltage is 1v peak to peak before clipping.

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

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.

My Sanders sound pre has a gain setting I think of 8DB, or close.

   I set it at 3 DB, for a bit of boost, but the McCormack amp does the rest

SPL in your room, in your sitting position is also largely contributed to by the room itself.
There is a phenomenon called cabin gain or the "transfer function", I am going to quote from car audio information.

Have you ever wondered why car audio systems can have so much bass as compared to a home stereo system? The reason has to do with space. Your home has a lot of it and your car doesn’t. Because there is so little space in a car the bass notes (which are long pressure waves) build up inside the passenger area. To calculate the length of a sound wave you divide the speed of sound (in feet per second) by the frequency. At sea level, the speed of sound is approximately 1,127 ft/sec. For example, a 40 Hz note has a wavelength of approximately 28 ft (at sea level).

(speed of sound)/(frequency) = wavelength
OR
(1,127 ft/sec)/(40Hz) = 28.175 ft

Since the length of the average car interior (including trunk) is in the 12 ft range the 40 Hz note will be longer then the car’s interior.

This is why notes below 70-90 Hz (depending on the vehicle) will have a greater output than the rest of the frequencies. Once this magic frequency is reached, bass output will increase by about 12 dB/octave below that frequency. This phenomenon is called cabin gain or the "transfer function". So a smaller vehicle will have a greater cabin gain and should be able to have greater low bass than a larger vehicle. This is true for identical subwoofer systems with identical power.

The physics involved are true for listening rooms, and so SPL isn’t just about the amount of energy the drivers are feed or are producing, measured at the listening position. Many use control devices to attenuate and clean up low frequency nodes which can be attributed to cabin gain, or "transfer function". Boundary reinforcement is another interesting topic and is related.

I am sure it’s very well known by most of us, but often not hardly thought about. I may be wrong but it appears, MC takes this also into consideration?

It sounds goofy but at least it is clear I know the difference between volume level and SPL. Sound pressure level is what we measure with decibels. Literally the pressure difference between the compression and rarefactions of the sound wave.