feat: add comprehensive logging system with file output

- Add backend logging to logs/backend.log with file rotation support
- Add frontend logging to logs/frontend.log via /api/logs endpoint
- Add frontend logger utility with batching and user context
- Update .gitignore to exclude log files but preserve logs directory
- Update CLAUDE.md with logging documentation and usage examples

Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>

.gitignore

@@ -15,9 +15,11 @@ coverage
 *.lcov
 
 # logs
+logs/*.log
 logs
 _.log
 report.[0-9]_.[0-9]_.[0-9]_.[0-9]_.json
+!logs/.gitkeep
 
 # dotenv environment variable files
 .env