I grew up in the south and being a fairly warm climate, never had to chop wood or cut down trees. So, my education on chopping tools is fairly new and I am by no means an expert, but I have learned a few things that I would like to share.
First off, an ax is is not a felling ax and it's definitely not a splitting maul. "What's a splitting maul and why can't I use my utility ax to split wood or cut down a tree," you ask? Well, for starters you can, but I don't recommend it. An ax designed to cut down trees is designed to cut down trees, not split wood. A splitting maul is designed to split wood, not cut down trees. Your general purpose utility ax can do each of those, just very inefficiently.
Take a gander at this felling ax:
It has a very thin blade and weighs only about 3-4lbs. This is so that it can be hefted sideways and cut deeply against the grain. Some felling axes are double bladed and that's fine. Blades dull and two blades will last twice as long as one. The down side to the double bladed (double bit) ax is that the handle must be straight and may not be the most efficient swinger out there. This fiberglass handle from Northern Tool is more durable than wood and easier on the hands in the event of a splinter.
Next, here's a splitting maul:
This maul is heavy at 8-12 lbs and is designed to deliver a lot of downward force. The blade is fat and rounded to prevent the blade from sticking into the grain of a stubborn log. Generally, you will spit logs much faster with one of these than any other ax-type hand tool. You should never use the back of the maul as a sledge hammer or use it to drive in splitting wedges. The type of metal used in this tool is very brittle and can send off shards of metal causing injury.
Splitting wedges:
Just when you thought you had it figured out, there are different types of wedges too. Metal wedges are for splitting logs. Don't use them for anything else. Generally, if you are trying split some seasoned hardwood you will run into some that just won't split with the maul- that's when you use the wedge with a sledge. Start the cut with the maul and finish it off with the wedge.
Plastic wedges are used for felling trees. Never attempt to fell a tree with a metal wedge. Plastic ones are safer especially when you are dealing with a chain saw. So for big trees, you run the risk of pinching your saw blade between the cut in the tree- that's when a plastic wedge comes into play. you can use a hatchet or a splitting maul to drive one of these in- it really doesn't matter. When you use it with a chain saw it won't damage the chain if you accidently touch it. You could use them for splitting wood, but the metal ones are more durable.
Finally, my little stint here on explaining wood cutting tools is just the first chop at a 100yr old oak. There are LOTs of different types of tools and many varieties in between for different applications. Fireman's axes, hatchets, throwing hatchets, camping ones, killing ones... you get the idea.
So, the moral of the story is, if you only have one type of ax in your survival tool kit, you might want to check and see what kind it is and then go and get another one.
I recommend at least a felling ax and a splitting maul in addition to a camp hatchet for limbing. Also, some advice: when you buy an ax, it doesn't come sharp. You will NEED to sharpen it either with a hand sharpener or a bench grinder. Also, they don't stay sharp- be prepared to sharpen your tools (including shovels) with hand files and stones.