I have logged a complaint with your customer care desk.

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Only the second seems reasonable.
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Thank you for the reply, Mr Micawber.

Mostly I have seen lodged being used in the context of complaint. But some days ago I read something like: Dear Customer, Your complaint logged with our customer service....

I believe the definition given below has some room to be used in the context of complaint but I would still say using lodge is more idiomatic. Please let m

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Now you have presented an example in the passive voice, and the company is saying that it has written down a memo of the complaint. This is probably more common than I would like, since it suggests to me that the company has noted the complaint but possibly done nothing to resolve it.
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I had the same dilemma earlier today. Now after looking up various examples, I think that "lodge a complaint" means to file a complaint, whereas "log a complaint" means to add it to a list.

If somebody is phoning up a Complaints Department, I would suggest it is more accurate for them to say "I would like to lodge a complaint" - OR "I would like YOU to log a complaint" as the person calling is doing the "lodging" and the person taking the call is doing the "logging".

Jackson6612I have logged a complaint with your customer care desk.I have lodged a complaint with your customer care desk.