Based on your description this information is available only within your process. The recommended approach would be to leverage your logging framework and populate the log that you write with required information.

Consider using structured logging to print the formatted log records to STDOUT. Cloud Run employs an implicit logging agent which can parse lines of logs printed to STDOUT (or STDERR) and ingest them as logs with structured (JSON) payloads. Look at example here: https://github.com/minherz/o11y-methods-screen-demo