My first variable power scope was a Leupold 3-9X that I had mounted on a Ruger 77--my first "real" gun. After I mounted it I cranked it up to 9X and don't think I turned it down ever that I can recall.
It was reading about Africa and hunting dangerous game--long before I ever had a chance to visit the Dark Continent--that I learned about the virtues of hunting with scopes set to lower power, a philosophy that is second nature to me today. The vast majority of my big game kills are at 4X or so. In the case of longer shots, off a steady rest, I might go to 8X but that is a rare exception.
The benefit of using a lower magnification setting a simple: you get a better field of view, making it easier to make follow-up shots; the exit pupil is larger, making it easier to get a good view of whatever you're looking at through the scope; lower power settings reduce the sense of crosshair tremor and wobble, improving shot placement; and, you don't really give up anything in the way of accuracy. Trust your zero, put the crosshairs where they need to go and the bullet will do its job.
So, if my big game "sweet spot" is 4X, what's yours? How far are your shots and what do you set your scope at?