[Misc] add structure logging, write to file and log tracing for SGL Router

sgl-router/README.md

@@ -67,6 +67,20 @@ $ pip install -e .
 
 **Note:** When modifying Rust code, you must rebuild the wheel for changes to take effect.
 
+### Logging
+
+The SGL Router includes structured logging with console output by default. To enable log files:
+
+```python
+# Enable file logging when creating a router
+router = Router(
+    worker_urls=["http://worker1:8000", "http://worker2:8000"],
+    log_dir="./logs"  # Daily log files will be created here
+)
+```
+
+Use the `--verbose` flag with the CLI for more detailed logs.
+
 ### Troubleshooting
 
 1. If rust analyzer is not working in VSCode, set `rust-analyzer.linkedProjects` to the absolute path of `Cargo.toml` in your repo. For example: