To understand this correctly... 8ohms has more impedance thus lower wattage...
That's generally correct. If you cut the impedance in half, you get twice the power.
In order to understand that you first need to know that when operating normally, most amplifiers are "voltage limited", and like most electronics, they are "constant voltage"* devices.
Related - If you put two 8-Ohm speakers in parallel, that's 4 Ohms total... If you have one speaker running at 3W, and you stick a 2nd speaker in parallel, they both see the same voltage, they are both running at 3W and you have twice the total power (as long as the amplifier holds-up).
Wattage ratings on a speaker are a little complicated... A 3W speaker is designed to handle a 3W amplifier operating at 3W peak with undistorted program material. Regular program material (voice or music) usually has an average power of about 1/10th the peak power... So, a 3W amp (at full undistorted volume) is putting-out about 0.3W average and the speaker can handle that. You could probably burn-out the 3W speaker with constant full-volume test tones, or with a distorted guitar.
Or, if you use an 8W amp with a 3W speaker and play it a full volume, you could burn-out the speaker. But as long as you don't "blast" the volume, you should be OK.
You won't get 8W from a 9V battery. An AC waveform swings positive & negative, which means [theoretically[/b] you could get +4.5V and -4.5V peak. But, an AC waveform only hits the peaks for an instant, and the RMS (a kind of average) is about .707 times the peak. If you could get the full 9V peak-to-peak voltage swing, that's about 3.18V RMS and you could get about 2.5 Watts into 4 Ohms. ...But, you'll get some "drop" through the amplifier so you won't get the full voltage, and the 9V battery can't supply that much power (voltage and current at the same time) and you'd be lucky to get 1W.
...less decibel range than the 4ohm 3 watts?
Only if the speakers have equal efficiency. If the spec is published, speakers are usually rated for a dBSPL rating with 1W at a distance of 1 Meter. (Technically, that's sensitivity, not efficiency, but they are directly related.)
Ohm's Law and Power formulas:
[u]Ohm's Law[/u] defines the relationship between voltage, resistance, and current. Basically it says more voltage = more current, and more resistance = less current. (Resistance is the resistance to current flow).
(Impedance and Resistance are both measured in Ohms in some cases (like with speakers) they can be considered the same.)
Current (Amps) = Voltage/Resistance
With a little algebra, you can get:
Voltage = Current x Resistance
Resistance = Voltage/Current
The basic power formula is:
Power (Watts) = Voltage x Current
And, if you know Ohm's Law, you can get:
P = V2/Impedance (This one is handy for amplifiers & speakers.)
P = I2 x Resistance
