If I understand your question, SmartThings already does this automatically for every device attached to the network. It logs every event and the logs are maintained for seven days. If you just want to review the logs you can open your phone anywhere and see the log for that specific device.
So basically anything that happens on your SmartThings network is already recorded to the logs for your SmartThings cloud account, and you can access that many different ways. But the data is only kept for seven days so if you want it for longer than that you have to save it yourself somewhere.
Here’s the log from one of my motion sensors. This is just a screenshot from the official SmartThings mobile app.
And this FAQ discusses the various different types of logs that are available:
You can also collect the information in various ways and use it for other things. A lot of people use grovestreams. But, also as @theedpope suggested, you can just use IFTTT to log it to something.
