How can I correctly capitalize this e-mail title: Unable to Log In to System
For bonus points, what are the parts of speech for the short words? I'm having particular difficulty with the word "to".
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I'd use sentence case:
Unable to log in to system
(Actually, I prefer "log into", but see https://english.stackexchange.com/questions/5302/log-in-to-or-log-into-or-login-to - where the top-voted answer favours "in to" but admits that "into" is probably just as widely used.)
The rules for title case vary slightly among different style guides. It could be:
Unable to Log In to System
Lower case is usual for the shortest words ( http://blog.apastyle.org/apastyle/2012/03/title-case-and-sentence-case-capitalization-in-apa-style.html ), which would point to "in". However, although normally a preposition, "in" here is part of a phrasal verb and is behaving adverbially, so Chicago style at least would recommend "In" ( https://english.stackexchange.com/questions/169648/how-to-capitalize-get-out-of-in-a-title-according-to-cmos/170829#170829 ). I think that "Log In to System" looks slightly better than "Log in to System", although I dislike both.
You also asked about the parts of speech. You can find these in a dictionary. Traditionally, both "to"s would be classed as prepositions, although the first "to" is really better regarded as an infinitive-marking particle (as is now recognised by Oxford: https://en.oxforddictionaries.com/definition/to ).