I don’t understand why you would have a logging microservice vs just having a library that provides logging that is used wherever you need logging.
Logs need to go somewhere to be collected, viewed, etc. You might outsource that, but if you don't it's a service of it's own (probably actually a collection of microservices, ingestion, a web server to view them, etc)
Only good reason would be for bulk log searching, but a lot of cloud providers will already capture and aggregate and let you query logs, or there are good third party services that do this.
Pretty handy to search a debug_request_id or something and be able to see every log across all services related to a request.