I've got a kukri, some hatchets a real axe, saws, many knives, etc. If I had to pick a combination of them on which to rely for every day stuff, not specific tasks, and I could not simply be stationary and have a collection of tools, the axe would be left behind. I'd bring the kukri machete, my multi-tool, and a pruning saw.

A small coldsteel kukri weighs 60% or less of the wight of an axe, and I can stuff the whole thing into my BOB. Beats walking around in a populated area with an axe. I have a small folding pruning axe with which I have taken down some small trees, and it weighs ounces. Between the two, I think I could handle a lot of tasks.

When I consider picking a few things and think those would be the ones, should I ever need to bug out, the tools being light and able to handle a number of tasks is key.

The portable chain saw mentioned by Bejammin sounds like it might be something to try.