I actually discovered the remedy for logs stuck in cliffs(and walls) back in world 4, but I never really shared it with anyone but a few of our villagers and it seems to still be an issue from what I've seen on the forum, so here I present you with my formula.
What you need to do:
1. Get a wagon.
2. Ride it in the first seat and remain in it while carrying out the other stages, second seat might be able to do it as well.
3. Ride as close as you can to the log you wish to remove.*
4. Select "lift" and click on the log.
5. The log should now be removed from its place and be lifted above your head.
* I don't remember if you had to put your character in the wagon close to the log or if it was the cattle pulling the wagon that had to be close to the log.
---
This worked in world 4 at least, I did it myself several times, but I do not currently posses a wagon so I hope some one else can prove this for me.