Hi,
Have you checked that all the power saving options available to you are set to the values that give you the best chance of maximizing the battery life.
Ensure that the Power Saving (Auto Power Down) setting is set to '''on'''.
http://www.manualslib.com/manual/220667/Canon-Powershot-Sx120-Is.html?page=117#manual
Ensure that the screen shut off time is set to <1 minute
http://www.manualslib.com/manual/220667/Canon-Powershot-Sx120-Is.html?page=118#manual