I can only find the medium base replacement lamp. The base I need is larger. Is the bulb shown here the one I need?

By base, do you mean the actual part of the lightbulb that goes into the socket? If so, This is a mogul style base. It is mainly used in the flourex floodlights and is specifically designed so that the ballast is in the fixtures themselves and not in the lightbulbs. This is not a standard light socket size and would not work in a normal indoor light socket

Yes, this bulb has the larger medium base, which is very hard to locate in a floursecent bulb.

yes, if this is really a "mogul" base bulb. You'll have to check at the store to see for sure.