This is a docker issue. I'd suggest you set up a logging driver. From a brief look it seems like this is a runtime option.

More details: https://docs.docker.com/config/containers/logging/configure/ and https://docs.docker.com/config/containers/logging/

FusionAuth does not have any plugins that directly write to 3rd party services like DataDog.

But with the API https://fusionauth.io/docs/v1/tech/apis/event-logs you can write an ingester.

Per this issue https://github.com/FusionAuth/fusionauth-issues/issues/673

"This is because the services are running interactively and the logging is written to to stdout."

So when you start up your fusionauth instance using startup.bat a cmd window will open up and your logs will be written to that.