OK, so now it's the next day, and I've got some results.
AS it turns out, and I should have known this, the base weapon damage is irrelevant. What matters is the attack roll - BAB, Str mod, and weapon attack bonus vs. target AC - and any bonus to the damage roll - weapon attack mod and special feature. Since you didn't provide the target's AC, I've assumed 25.
I computed the average damage considering hit chance (but I chose to neglect crits for simplicity) for weapon bonuses from 0 to +5, both with and without the Dominance enchantment, using the BAB of 10, the Str mod. of 5, and the greataxe as directed. And here are the results.
Table 1: Damage With and Without Dominance
Weapon Bonus |
W/o Dominance |
W/Dominance |
Delta |
0 |
6.325 |
7.55 |
1.225 |
1 |
7.5 |
9.075 |
1.575 |
2 |
8.775 |
10.7 |
1.925 |
3 |
10.15 |
12.425 |
2.275 |
4 |
11.625 |
14.25 |
2.625 |
5 |
13.2 |
16.35 |
3.15 |
(6) |
(14.875) |
|
|
(7) |
(16.65) |
|
|
Average |
|
|
2.129167 |
(I included +6 and +7 for a reason that will soon be clear.)
The average increase given by the dominance feature is a bit over 2. That is similar to the improvement in average damage given by a +2 weapon, actually 2.45. So I would be inclined to call dominance a +2 enhancement. But wait; there's more.
Another way to approach this is to consider how large a weapon attack bonus would be needed to achieve the same average damage as dominance. In other words, for a weapon with a +n attack bonus and dominance, what +m would I need without dominance to get near to the same damage.
Table 2: +m for Nearest to Equivalent Damage
Weapon Bonus (+n) |
Dam W/Dominance |
Nearest Bonus W/o (+m) |
0 |
7.55 |
1 |
1 |
9.075 |
2 |
2 |
10.7 |
3 |
3 |
12.425 |
5 |
4 |
14.25 |
6 |
5 |
16.35 |
7 |
So, for an underlying weapon bonus of 0, 1, or 2, dominance gives extra damage as an enchantment that's one better, i.e. a +1 enchantment. But for an underlying bonus of 3, 4, or 5, it's +2.
Bottom line: all in all, I'd call it +2.