I also need this functionality. A simple download button on the events page would be so simple!
I was able to extract the entire event log by manually calling the API.
I used the chrome web inspector to see the API address and copied the URL. I changed the length value (near the end) to 999999. This produced JSON with all events. A quick paste into notepad++, replace all ],[ with \n gave me a nice list. With a few more quick modifications I have a list ready to paste into excel. You would think you could use the url below by replacing the IP address and device number.