Use CreateExportTask com um AWS SDK

    using System;
    using System.Threading.Tasks;
    using Amazon.CloudWatchLogs;
    using Amazon.CloudWatchLogs.Model;

    /// <summary>
    /// Shows how to create an Export Task to export the contents of the Amazon
    /// CloudWatch Logs to the specified Amazon Simple Storage Service (Amazon S3)
    /// bucket.
    /// </summary>
    public class CreateExportTask
    {
        public static async Task Main()
        {
            // This client object will be associated with the same AWS Region
            // as the default user on this system. If you need to use a
            // different AWS Region, pass it as a parameter to the client
            // constructor.
            var client = new AmazonCloudWatchLogsClient();
            string taskName = "export-task-example";
            string logGroupName = "cloudwatchlogs-example-loggroup";
            string destination = "amzn-s3-demo-bucket";
            var fromTime = 1437584472382;
            var toTime = 1437584472833;

            var request = new CreateExportTaskRequest
            {
                From = fromTime,
                To = toTime,
                TaskName = taskName,
                LogGroupName = logGroupName,
                Destination = destination,
            };

            var response = await client.CreateExportTaskAsync(request);

            if (response.HttpStatusCode == System.Net.HttpStatusCode.OK)
            {
                Console.WriteLine($"The task, {taskName} with ID: " +
                                  $"{response.TaskId} has been created successfully.");
            }
        }
    }