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If you can use the particle (in this case, "on" as an adverb) with the verb
alone, it's part of the the verb rather than a preposition relating a verb
to an object.
Since "log on" can be used intransitively as an imperative statement, we can
safely assume that "on" is actually part of the verb phrase.
EXAMPLE
Sit down at the terminal and log on.
Here's what AHD (online ed.) says about the use of "on" versus "onto":
In constructions where "on" is an adverb attached to a verb, it should
not be joined with "to" to form the single word "onto": "move on to
(not onto) new subjects"; "hold on to (not onto) our gains."
Logging off,
Bill Burns *
Assm. Technical Writer/Editor * LIBERTY, n. One of imagination's most
Micron Technology, Inc. * precious possessions.
Boise, ID *
WBURNS -at- VAX -dot- MICRON -dot- COM * Ambrose Bierce
