When it’s time to start work you probably “log on” to your computer using your “logon” or “log-on.”
The latter is a noun, the former a verb, and they have not yet fused into one word for both forms.
Here’s what The Associated Press Stylebook says:
“Login, logon, logoff (n.) But use as two words in verb form: I log in to my computer.”
Words are often created when someone takes one form and starts using it another way. This is referred to as a “back-formation.” Sometimes, the noun leads the way. For example, the word “target” was acceptable only as a noun as recently as the first edition of the 1964 Webster’s New World College Dictionary. In the case of “log on,” the verb came before the noun.
Although “on” is often a preposition, in the verb “log on,” it’s an adverb.
You wouldn’t say, “I’m going to log the computer” unless you’re putting the computer on a list. The “on” is necessary to further explain the action. Another way to better understand this is to make the verb another tense. Today you “log on,” but yesterday you “logged on.” No one is suggesting that this become “loggedon.”