Originally Posted by a mere housewife

Well 'logon' is a variant of 'login' which means 'to identify oneself to a computer by entering username and password'. 'Log' is less precise it seems (for the purpose of identifying oneself), meaning 'to enter into a log, to keep a record of'. So I guess if the emphasis is on the record keeping of your login then you ought to use 'log onto' or 'log on to' (I would use 'onto' rather than 'on to' because it is more efficient, and why use two words when you can smush together?) but if the emphasis is on your identifying yourself to a computer you should use 'logon' or 'login'. If the emphasis is that you felled trees on a server then I would suggest moving the server.