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You can send the logs to CloudWatch Logs, then into Amazon's hosted ElasticSearch service:

http://docs.aws.amazon.com/elasticbeanstalk/latest/dg/AWSHow... http://docs.aws.amazon.com/AmazonCloudWatch/latest/Developer...

Use an external logging service. You can run your own if you like, but there are plenty of people that offer pretty good services for not much- assuming you're not generating more than a GB or 2 a day.

I have personal experience with Papertrail and Loggly, which is quite nice if you want to build derivative data.

Yes, it isn't great .. you ship the logs to a bucket or you use an external service. We ended up doing both.