Command Block Help (CanDestroy Command)?

My concern is, how can I obtain an axe to put in my world that can only break logs? Like in the item description it would say: Diamond Axe, Can Break: Logs. That way in adventure mode they can only use it to break logs. I'm very new to this so sorry if this is obvious.

Added (1). Someone on the minecraft fourms tried helping me and suggested the /give @p minecraft: diamond_axe 1 {CanDestroy:["minecraft: log"]} I tried this and it says that {CanDestroy:["minecraft: log"]} is not a valid number. Anyone have any ideas?

That would require a mod affecting the way axes work.

No it wouldn't
try:
/give @p diamond_axe 1 0 {CanDestroy:{log}}