Q:

How to know which key a song is written in with the same key signature?

How to know which key a song is written in with the same key signature ? For example, the C Lydian scale has the same key signature as G major (G Ionian) AND E (natural) minor (E Aeolian) which makes it pretty difficult. Sometimes you can tell if in the bass staff, there are chords in a specific key, for example  C major: CEG - CEG. But this isn't always the case. So how do you do it?

A:

With no key sig, the piece could be in one of several keys. C maj., and A minor being the most likely. What clues are there? In Am, there will likely be some G# notes. 
However, it may also be using D Dorian, E Phrygian, F Lydian, G Mixolydian, or, unlikely, B Locrian. The melody would centre around the appropriate note, and would feel at rest on that note. For example if it was D Dorian, the last chord may well be D minor, and last note D.
That apart, there's no way of telling - unless it's announced at the top! The accompanying chords will all come from the same set, with perhaps the additional E in Am, which is where the G# came from. Otherwise, it's usually academic.

