Tracking Potential Bathtub Leak Without Removing It

I've noticed that the corner of my bathtub gets wet. When a paper towel is placed there, it gets soaked in a short while. There is no large amount of water leaking out though. That spot just gets moist especially after the bathtub is used.

Is there a way to trace the leak without pulling out the bathtub?

(Note there's a wet spot on the baseboard molding and I don't know how water got there either.)

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