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Right, that’s another way of do it. You keep with your left hand the ax on the wood and with your right hand you hammer the mallet on the ax, what kind of right mallet for this kind of job would be best suitable? I also guess that if you’re not exactly inclined to manual work that it may result a bit tricky.
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It's useful for bigger logs if you've not got a bigger proper wood splitting axe and the hatchet isn't getting through. Or If you want an accurate cut and you're not accurate swinging. Wooden mallet, or a big log also works.
Not sure the specifics, bit my fancy hultafors specifically says not to, whilst my cheap one has a reinforced 'back' and works nicely. It's definitely more wedgy in shape than the nice one, which is very thin. But then i use it mostly for shaping and carving than splitting.
If you're going to potentially be using a wooden mallet on the back, make sure you get one with the right head.