Please note that a mission can be completed only if you have performed the action that was requested after the mission appeared.
We understand that sometimes players tend to clear the island and perform some actions before the game asks them to do so. In this case you just need to wait until another dry log appears on your Home Island and cut it down.
Don’t worry, the dry tree will definitely appear. By the way, this mission doesn’t stop your progress, so you will complete it sooner or later.