No, not really, in general, a battle axe is really well balanced and tend to weight less than a similar lenght sword, wood could be more resistant and ligther than most form of pre-industrial iron, and the fact a battle is weigthed in both extremes, in the axe head proper and with a counterbalnce near the handle, making his use as fast a sword, and with more weight in the strike zone that coud mean you do blunt damage even againts a heavily armored oponet,
now if your are thinking ina typical utily axe or woodsman´s axe you are right they are cumberstone and exhausting to use in battle, but they aren´t designated or balanced to be used in battle, but will do in a pinch