I noticed tonight the wall in my son's closet seemed damp. I cut a piece of the dry wall out in order to see where the moisture was coming from. His closet is adjacent to the bathtub. Underneath the bathtub you could see signs of water damage however there was no water. I ran the shower for 10 minutes but could not reproduce a leak. I then ran the facet for 5 minutes and no water damage but the tub began to fill up as soon as I closed the drain to trap the water it began leaking immediately and the water under the bathtub began collecting on the floor. As soon as I opened the drain the leaking stopped and the water started to drain out from the floor. I could not see where the water was coming from only that it was coming from the drain side of the bath tub and the water began to go away after I opened the drain. I have a drop in tub and shower. I have a crawl space under the house so I crawled under the house and seen no signs of damage or leaks from underneath the house. I assume I have some sort of problem with the drain I am just not sure what it is and would not like to cut a hole in the other closet on the other side of the tub where there is no signs of water damage to figure out where it is leaking from.

Fill the tub past the drain in the bottom of the tub and close the stopper. If it leaks then it is probably the tub drain seal. Most tub drains can be tightened by inserting two screwdrivers (they do have a tool for that though and I recommend using it) into the drain from the tub side and twisting them clockwise. The goal is to tighten the drain down onto the seal. Hopefully that works or you will have to replace the seal.

If the tub leaks when you fill the drain past the overflow (the metal part near the top of the tub) then the overflow will have to be tightened.

If your really unsure about all this try a handyman from Craigslist. A plumber will do the job right but you will also have to paypremium for that.

I closed the drain to trap the water it began leaking immediately
Mustomper nailed it! You have a ruptured drain seal. This is a easy fix for the do it yourselfer.
You will first need a tub tool,(see image). A pair of pliers with the handles down and a screwdriver for leverage works also. If you have a trip lever drain then loo down at the water drain, See the cross bares in the drain? Take 5the tub tool and turn the drain out counterclockwise.
Now clean all the old putty from the tub and tub drain.
Make a small rope out of Silicon Jel or Plumbers Putty and run it around the lip of the drain and screw it back inn, tightening the lip as the seal squeezes out. MAKE SURE IT'S ALL OUT AND THE DRAIN'S TIGHT!
Clean up and enjoy a leak free rub-a-dub. Good Luck, Tom