To lodge a complaint is to make an objection through the proper channel, so that it exists on paper and somebody has to answer it.
It takes with and against: lodge a complaint with the council against a neighbour. Lodge means to put a thing somewhere it will stay, which is exactly the point — grumbling to a friend is not this, and the formality is the whole meaning. File is the American preference and make is the plain one, so use lodge when the letterhead matters and make when it does not.
1phraselodge a complaintTo make an objection through the proper channel.
“If the noise continues, lodge a complaint with the council.”
synonyms: file a grievance
lodge a complaint is a formal objection.
Also accepted in games: make a complaint
Practised in Common Word Pairs