Lumberjacking is the Skill used to acquire logs out of trees. Double click a Hatchet and click a tree and after a few seconds you may get some logs in your pack.

Wood Skill
Log 0.0%
Elven Log 55.0%
Redwood Log 65.0%
Eldar Log 75.0%
Blackwood Log 85.0%
Lightwood Log 101.0%

Note that when you find a tree with one of the special logs (any other than Log) a Guardian of that type will apear. In order to get the logs you have to kill the Guardian, a Eldar Bow with some decent Archery skills is advisable.