If you know what champion came out when you joined, you can figure out a date range, but there's nothing to give you an exact date.

-Banister

Why do you need to know the exact date you joined? and yah you go back by events.

Checking E-mail is probably the easiest way to find out... My join date was two years ago but was able to dig up the E-mail no problem.

Yeah you should check your email if you use gmail you can easily type in riot or league of legends in the search box and it pops up when you signed up and it will tell you the exact time of day you got the email too. If you have yahoo or any other mail provider I think it still will say something in there about when you joined.