My neighbor is directly below me. You cannot hear dripping unless I plug the tub and fill it a little and then let it drain. I have a soaking tub so it doesn't even come close to the over flow. There is no water damage below. When you just let the water run full without plugging the tub you cannot hear the dripping below. Only when I fill it and then drain it. There is no access unless I tear out a ceiling or wall.

First, don't start a new thread to continue the discussion. Just make your response in the "Answer this question" block.

If the dripping is heard only when you pull the plug and let the tub drain, and you are not causing the tub to overflow, try this.
Partially fill the tub and let it sit for a little while. See if neighbor hears dripping. If dripping is heard then leak is around tub flange.
If no dripping is heard, open drain and let tub drain. If dripping is heard then the leak is in the drain pipe. It is conceivable, but would be unusual, to have a leak that only leaks when the large volume of water from draining the tub is present but does not leak when lesser volume from the faucet is present.
If you get indications that the tub flange is leaking it is possible to tighten the flange from the tub side. If you get indications that the drain pipe is leaking and you have no access panel, you will have not choice but to open the ceiling below to repair. Sooner or later the leak will cause damage to the ceiling below and repairs will have to be done anyway.

PS Examine tub carefully. It is possible that you have a cracked tub (assuming acrylic tub).