From Nookipedia, the Animal Crossing wiki
Axes are a type of tool that can chop down trees when players are standing next to and facing them. The regular axe is the easiest axe to obtain, by going to Tom Nook's Store while the other two, the silver and gold axe, are harder to obtain. After a regular axe is used against something a number of times, the axe will shatter. This will be apparent when the image of the axe in the inventory appears to have a crack in the blade. However, silver and golden axes do not break and can chop down trees faster than normal axes. They can all be used on money rocks as a substitute to the regular shovel, but this will also cause regular axes to break quickly.
 In Animal Crossing
 In Wild World
Purchased from Tom Nook's Store for 400 Bells. This axe will break after chopping down a few trees.
Players obtain the golden axe by completing the Golden Axe Trading Event. This axe will never break.
 In City Folk
Players obtain a regular axe by purchasing it from Tom Nook's Store for 400 Bells. This axe will break after chopping down a few trees.
The player needs to throw an axe into the town fountain (obtained through donation to the Town Fund). Once the axe is thrown in, they will have to sweet-talk Serena in hopes of getting one, but they can try only once per day. This axe will never break, unlike the normal axe.
It is obtained the same way as the silver axe, but it is harder to get. Like the silver axe, the golden axe will never break, but it will chop down trees much faster than the other two axes.
 In New Leaf
Players purchase the silver axe from Grams for 8 Medals on Tortimer Island. It is a rare find at the island. The silver axe can still break, but it lasts longer than a regular axe and always creates a rare stump pattern.
Players can obtain a gold axe from Leif at his shop after buying at least 50 tree saplings. It is unbreakable, cuts faster, and creates a rare stump pattern occasionally.
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