How to create Google Cloud Logging entries that are correlated with the request log's trace ID?

Using the Google Cloud Logging client library for go, it's possible to specify the Trace field on an Entry struct: https://cloud.google.com/logging/docs/reference/libraries#log-entries. By default, on Cloud Run, requests are logged with a trace ID. How can I get this trace ID and specify it in the logs I create so I can correlate logs by the request that caused them?

Requests that come in to Cloud Run will have the header X-Cloud-Trace-Context. This header is actually traceID/anotherID. When you create a logs Entry with the cloud.google.com/go/logging package, you need to specify the Trace field in the format projects/{$project_id}/traces/{$trace_id} with your project ID and the trace ID from the header.

In my code, I get the trace ID from the request header and set it on the request context c like:

ctx := context.WithValue(c.Context(), logging.TraceId{}, c.Get("X-Cloud-Trace-Context"))

where logging is my own custom logging package. Then inside logging code, I have this method to attach a trace from a context to a Cloud logging entry:

func populateLogsEntryWithTrace(ctx context.Context, entry logging.Entry) logging.Entry {
	traceID, ok := ctx.Value(TraceId{}).(string)
	if ok {
		trace := strings.Split(traceID, "/")[0]
		entry.Trace = fmt.Sprintf("projects/%s/traces/%s", projectID, trace)
	}
	return entry
}