To log is to record. To lodge is to push something in so it stays there.
While sorry to be a party pooper, I have to say that both log and lodge are actions a user cannot perform to the ticket, only the Help Desk can. What users do is to open a ticket.
- You have a "kind of ticketing system to track reported issues."
- You have the system at the Help Desk.
- You track the issues by logging them based on the problem they have reported.
You ask people to file a ticket by saying "Please open a new ticket and log your issue. Label it ABC, as that is the category you described. We will look into it based on its priority, and follow it through to resolution within the next 24 hours."