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Monitoring and troubleshooting applications running in Docker containers is essential for maintaining the health and performance of the applications. Viewing the container logs is a critical method for achieving this. When applications run within a Docker container, their output, including error messages, warnings, and other relevant information, is typically directed to the container's standard output (stdout) and standard error (stderr).

By using commands such as `docker logs [container_name]`, you can access these logs in real-time or review historical log entries. This log data is invaluable for identifying issues, understanding application behavior, and diagnosing errors that may occur during the runtime of the application.

The other methods mentioned have their own purposes but do not directly address the monitoring and troubleshooting of running applications as effectively as viewing container logs. For example, while inspecting images can provide information about the structure and layers of the image, it does not provide real-time feedback on the application's performance or issues occurring within the container. Likewise, using Docker Hub is primarily for image storage and sharing, and running diagnostic commands is more focused on checking the Docker environment rather than directly troubleshooting a specific application within a container.