American Heritage backs me up on this one: “log on” = verb, “logon” = noun.
@merriamwebster appears to claim that “log-in” and “log-on” are the noun forms, which is…preposterous, I think? Is this common usage and I’m missing something?! https://www.merriam-webster.com/dictionary/log%20on
(M-W also claims that “log on” is a synonym for “log in,” and that the former is more common. I’m…skeptical of both. I associate “log on” with coming online / connecting, and “log in” with authentication.)