You can now publish logs from your Amazon RDS for SQL Server database to CloudWatch Logs. Supported logs include both agent logs and error Logs. Publishing these logs to CloudWatch allows you to maintain continuous visibility into database errors and activity. For example, Customers can set up CloudWatch Alarms to notify them on frequent restarts that are recorded in the error log. Similarly, Customers can create alarms for errors or warnings recorded in SQL Server Agent logs related to their SQL agent jobs.
from Recent Announcements: https://aws.amazon.com/about-aws/whats-new/2019/11/now-publish-log-files-amazon-rds-sql-server-amazon-cloudwatch/