I have been playing the game since January, and am loving it! I have unlocked almost every gold tool, and I only need to unlock the axe. Problem is, I don't know how to unlock it. I've tried cutting down some trees with the silver axe but it hasn't unlocked yet. What do you have to do to unlock it?

You're on the right track. You're also an anomaly if you got the gold watering can before the gold axe. That watering can seems to take forever!

How many trees is required to unlock it? I've chopped 50 trees with my silver axe but nothing yet...
By the way, to unlock the gold watering can, you have to donate a good portion of the plants that you can grow. Should be easy if you got the update where they changed the plant breeding.

I wonder if it's the number of axes you go through? I chopped about 60, but I used a lot of brass axes because I couldn't afford silver. I'd buy a brass one every time I had an extra $100, and a silver when I could afford to spare the extra money. It was a low priority at the time.

I think it's related to the number of cut trees. It's most likely 50 trees because I counted and I cut exactly 50. Today I finally got the golden axe ^__^

You don't have to count. There are stats in the Mayor's office for both all mayors (sit at the table) and each individual mayor ("look" at the portrait). I think I know kind of vaguely what you need to do for each tool but I don't know numbers because I just don't care that much, lol.