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

The relationship between dB and power is logarithmic.
End of story.

This site is cries out for a Audio 101 list of stuff every audiophile should know. This would come right after W=VA.

They're not paying me enough. You could mention that.....

Power.  

Seriously. Helps to think things through. What is power? Current? Voltage?  
Here I'll give you a hint: Watt is power? That's not a typo. 

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.

This is nothing more than an unnecessarily techno-jargon laced way of saying exactly what I said: Gain is a multiplier, power is a ceiling.   

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.

Thanks everyone. I like the gas pedal vs engine analogy. But does a higher initial gain ultimately mean anything for the output of the amplifier (like higher possible SPL in dB for a given speaker)?

Yeah I was afraid of that, as it is a horrible analogy. So awful in fact I am gonna leave it alone and just explain what gain actually is, in the hopes the truth will replace all the false ideas generated by that awful example.

Gain is nothing more than another word for multiplier. Gain is the amount by which the incoming signal is multiplied. Because gain is a multiplier it is expressed in dB. So in your case, say you have selected 23dB gain. Whatever voltage level is input, is increased by 23dB.

Remember, dB is a log function. Remember, every 3dB is twice the power. Every 10 dB is ten times the power. So your 23dB equates to 200 times. (The math in this case is straightforward. But if you need help, ask!) Therefore, say your input voltage is .1mV. With 23dB gain your output voltage will be 20mV.

Flip the switch to increase the gain to 29 and since 29 is 6dB more than 23, that is two 3dB doublings, the output will be 4X the previous example or 80mV.

That is gain. It is like a calculator that no matter what number you enter all it does is multiply by 8. Or 16. Or whatever.

However, what if the calculator has a display with only 2 digits, and you input 100? Well clearly the gain does not change but the output cannot go that high, you are not going to magically get three digits no matter what the gain.

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.

Yes, and thanks.

The relationship is logarithmic. For every increase of 3dB power doubles. For 10dB power increases ten times. 

In your case switching between 23 and 29dB of gain is 6dB. So whatever input it takes to drive the amp to full power at 23dB, it will reach full power with 1/4 that level when set at 29dB.