If you know what part of the wall the pipes are running through, you need a space heater sitting near that area of the wall to help warm up the pipes. Be careful not to place it too close to the wall. Check on it often to make sure nothing is over heating.
Basically, you need to make the bathroom so warm that the pipes inside the wall will thaw out. Once they do thaw out, leaving a facet dripping will keep them thawed. If your sink has a vanity, open the doors under the sink so that the pipes under the sink are exposed to the heat also. If the frozen point is near that area, it will help it warm up.
The exposed pipe that is sticking out in the shower is not the problem.
Using a hair dryer will not help since you would have no idea of an exact point on the wall to point it at. The space heater would work if you just know which wall and a general area. It doesn't need to be as specific as a hair dryer needs to be.
The other problem is that the frozen spot in the pipe might not be in the bathroom area at all. Instead, it could be a frozen spot in one of the lines that goes to the bathroom from another point in the house. But, starting by warming up the bathroom is the first guess without knowing the layout of your home or the possible other points that the pipes might travel to get to that bathroom.