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The free tier has limited functionalities and reduced amount of data you can send them per day but should be enough for having fun still

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I was thinking about trying datadog for handling the logs. This looks interesting too.

I've used data dog before (for their free trial tshirt :D) it was pretty useful but I didn't get into the logs part of it, only the webserver monitoring.

You also have https://www.humio.com/ which also offers a free tier with 2GB of data per day and 7 days data retention. I was at their presentation at Yow! Conferences and the product looks good too. They also have a quite neat data compression strategy that chunks the data small enough that it can fit in L2 CPU cache in order to do decompression on the fly when searching a compressed DB.

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