I'd suggest using a logging crate, to reduce boilerplate code. Preferably one that uses log facade, such as env_logger or flexi_logger. I know flexi_logger can be customized a lot (formatting, where to write, mirroring to file and stdout), but other crates probably also have such functionalities.
This would also handle logs from libraries that are logging using log crate, which currently are being ignored (if there are such crates in your dependencies).