It's the second one. If you are merely asking someone whether he has logged on, then the "to" is of course not necessary. However, when you have an object, such as "the computer," you want to change "on" into "onto." To my knowledge, "on" and "to" are never used as separate prepositions. It functions the same way as "into." When "in" or "on" involve motion and an object, they become "into" and "onto." You wouldn't write "I am going in to the house" or "I am getting on to the roof." In the same way, "I am logging onto the computer" would be the correct usage.

__________________
DAVIDIVS DOCTVS VTRIVSQVE LINGVAE
Husband of Emilia
Member: First Reformed Presbyterian Church of Durham (RPCNA) - Durham, NC
Currently in the process of transferring membership to an as-yet-undecided church in Chapel Hill
Student: University of North Carolina at Chapel Hill, German Literature and Classics