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Hi and welcome to JustAnswer,
I would suspect either an infection or a tumor in the toe. Infections may be either of bacterial or fungal origin and take a long time to clear up. It may be necessary to remove to black nail and do a fine needle aspirate to look for bacteria or fungal spores to determine the cause and needed medications. However, your dog is old enough for this to be a tumor, though tumors don't respond to antibiotic therapy. Sometimes we just choose to remove the toe surgically to solve the problem without a long course of treatment and the possibility the the medications won't be enough. Here is more reading for you: