To break wood (logs) in adventure mode, you need a tool which is specially enabled to destroy logs by having it's CanDestroy tag set to include "minecraft:log" for oak, birch, spruce and jungle logs, and/or "minecraft:log2" for old oak or acacia logs. There is no way to limit it to specific data values, so you can't make an axe that is capable of chopping oak but not birch or something like that. The tag takes a list ([]) of item ids.
For example
/give @p minecraft:iron_axe 1 0 {CanDestroy:["minecraft:log","minecraft:log2"]}
gives you an iron axe capable of chopping all kinds of logs, which will be indicated in it's tooltip as well. It does not chop planks or any other wooden things.
NB: The mining speed is then determined by the tool itself, so while you can make log-chopping sticks, the speed would be the same as hitting a tree with a stick in survival.
