The way to place the logs to create a path that allows the hero to access the exit door is shown below.
To show that this is possible, the calculations below are performed. Assuming that each log needs at least 8 cm from each side to be fixed steadily between two edges, we can easily find the length "x" in the figure below. In this case, the length "w" should be equal or less than the length "x", which turns out to be true.
Name : Munzer
Server : PTR X3 (ts80.travian.com)
Only the second door is safe.
Let's assume that the first door is a trap. It says both doors are safe, but since it's a trap the sign is false and at least one of the doors is it a trap.
If we assume that the second door is a trap too, then the sign on it that says "both doors are safe" should be true, which conflicts with our first assumption. When assume that it's safe the sign is false, and no conflict occurs.
Assuming any different scenario would create similar paradoxes, which makes this the only correct solution.