Docker containers can be logged using various drivers, including the awslogs driver, which facilitates logging to AWS CloudWatch for centralized management. This process involves setting up AWS with the necessary permissions, configuring Docker to use AWS credentials, and using the awslogs driver to send logs to CloudWatch. Once the logs are in CloudWatch, they can be exported to an S3 bucket for further analysis with systems such as the ELK Stack offered by Logz.io. The integration with Logz.io involves setting up a pipeline from CloudWatch to the ELK Stack using S3 batch export, allowing for enhanced monitoring and analysis of Docker container logs. Additionally, methods such as Kinesis and Lambda can automate the export process, offering alternative solutions for managing Docker logs.