To log on is the accurate phrasal verb we use when we begin to use a computer system by entering a password. To log off is to stop using a computer system.
You can also use the phrasal verb to sign in when you have an online account with a bank, a shop or an online service like spotify or gmail. You also sign in when you are a participant at a conference or an event. To sign in is when you write your name on a piece of paper.