What is logic/reason/rational thought? how can you prove that?
I would say that logic is to come to conclusions that are consistent with our "experience" of reality.
This can be personal experience. For example, I know that when I have knocked things off the table before they have fallen to the floor. I have also found that when an object hits a hard surface with some force, it may break. Therefore if I knock this cup off the table onto the hard floor I can logically expect it to fall to the floor and break.
I could also do simple experiments to quantify the "experience." I can envisage some simple experiments that should show that objects falling from the table would accelerate to the floor, and so I could draw a logical conclusion that an object dropped from a greater height would be more likely to break.
Alternatively, I could rely on other peoples' "experience." Sticking with the same example, if a number of people I trust tell me that they have knocked cups off the table and broken them I can conclude that if I do the same thing I will also break the cup. If a reliable source such as a peer-reviewed scientific study of cups falling from tables quantifies this, I can also use this to draw a logical conclusion.
In an ideal world, I would reproduce the other peoples' "experience" (especially with the "reliable" peer-reviewed study, where I am completely reliant on strangers. In practice I have to put some faith in them, either through lack of time/inclination, or practicality (I don't have a particle accelerator to hand to test the Standard Model of particle physics, for example). However, I can still refer to the peer-reviewed study to show that an argument is consistent with reality, and thus logical.