I am trying to understand the solution to a logic puzzle, but it confuses me on many levels
The problem is this:
And the solution is this: 
What confuses me is the following:
Why would it impossible for the prisoner to find the lady if Room 8 was empty?
The puzzle stated that Room 8 would contain the tiger. But it also stated that a room can only contain the tiger if it's sign was false. The sign of room 8 states that it contains a tiger. So if it has a tiger, doesn't that make the sign true, therefore showing it doesn't have a tiger?
Again, for room 9. The puzzle states that it has a tiger, even though the sign on it says that it contains a tiger. Since only a room with a false sign can contain a tiger, isn't that contradictory?
The puzzle states that the only way sign 3 can be wrong is if sign five is wrong and sign seven is right. I don't understand why this is. Can someone explain?
Thanks in advance.
If room 8 were empty, the sign on the door would give no extra information. See the answer to your next question for more detail about what we know if room 8 has a tiger.
The sign makes two claims it has a tiger and room ix is empty. For the sign to be true, both statements must be true, but for the sign to be false at least one of them must be false, so we know that "room ix is empty" must be false, since the room does have a tiger.
This is the same idea as 2. For an "and" statement to be false only 1 part of it needs to be false and it doesn't make the other part false.
3 says 5 is right or 7 is wrong, so the negation of this is 5 is wrong and 7 is right.