There are a couple of places a bathtub drain can leak. One is around the drain. The other place is where the vertical overflow tube and the horizontal drain come together.
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Leaking Drain Pipes
There are some compression fittings there, similar to the ones under your kitchen sink. You should be able to get to the compression fittings through and access door or from the basement.
Just like the sink drain fittings, these have washers in them. You can try to tighten them and it may stop the leak. Replace the washers when it will not stop leaking. They should be 1 1/2", she same as a kitchen drain. Use the adjustable pliers to loosen the nuts to replace the washers. You will have to pull the fittings apart to get the new washers on.
Tighten the nuts back up, snug but not too tight. Try running some water and see if you have solved your leak. Did you? Good you are done with this bathtub drain repair.
A leak around the drain is another matter. The strainer is held in place by a large nut on the bottom of the tub. The flange of the strainer is sealed with plumbers putty. There is also a rubber gasket between the big nut and the tub. Whenever you disturb this connection you have to put new plumbers putty around the flange.
They make a wrench that can loosen the strainer but the average person does not have one. The handles of metal pliers will work to loosen it if you use a screwdriver for torque. Loosen the strainer and clean the old putty away. Check the rubber gasket, if it is worn or broken, you should replace it. Put a new bed of plumbers putty around the drain and thread the parts back together.
Use the pliers and screwdriver to tighten the nut. The putty will ooze out, this is normal. Get the strainer good and tight. Then clean up the excess putty. Try running some water down and see if you have completed this bathtub drain repair.