A splitting axe is designed for splitting small log sections for firewood. It's not what you are looking for, since you will be banging a wedge to open up a log. Chopping a tree down with a splitting axe is also not to great, since the bevel is too steep for that, and the weight is too high. You'll benefit a lot more from a hunting axe for felling trees. Still, those axes are not suitable for splitting of logs. No axe is...you need a true splitting wedge for that, OR an old axe head you are willing to sacrifice for splitting. Since this wedge/axe head will be beaten up.
I would advice a small or large (depending on your budget) hunting axe for felling of trees. And a seperate wedge or old, el cheapo axe head for splitting.