Breaking multiple block types with one tool

I'm making a lumber tycoon in minecraft, and I have a command by the press of a button to take money from the player and give the product with the "privilege" to break logs, but it seems to only break oak logs.

The command I am using is: /give @p wooden_axe 1 0 ("minecraft:can_destroy":{"blocks":["log"]}}

How can I make it break all other types of log too?

You'll need to pick all of the other wood types: "log" defaults to "oak_log", since oak was the first tree type in the game.

/give @p wooden_axe 1 0 {"minecraft:can_destroy":{"blocks":["oak_log","spruce_log","birch_log","jungle_log","acacia_log","dark_oak_log"]}} should work.