How do I - Magnitudes of DSO's?

I'm a bit confused here, or maybe I'm really not, you can tell me.

Since it is nice and cloudy here, I was planning out my next viewing
sessions. I was looking at some of the Messier objects comimg up
(some neat Clusters!) and noticed some Galaxies.

"Hmmm...Magnitudes in the 9's. That's going to be tough," I think.

To check, I go back and look, at M81 and M82, the only two galaxies
I've found so far.

This is when things begin to go downhill.

According to what I found, M81 has a magnitude of 6.9 and M82 of 8.4.
Now, through my eyepiece, M82 looks MUCH brighter. Thinking about it
though, I realize that Andromeda has a magnitude in the high 3's but
it is definitely dimmer than 4th magnitude stars.

So, how do they figure the magnitude of a DSO? The nearest I can
guess is that they must take the total illumination over the entire
area of the object. That would tend to make M82 appear brighter in
the eyepiece because it is much smaller than M81, even though M81
would have the higher total brightness because it is so much larger.

This seems inherently confusing to me since the lower the magnitude,
the brighter the object should be, but I'm sure there is a reason for
it...

Thanks.

Clear, Dark Skies.

Mark