Not really as I don’t do anything like this and your end goal is unclear. Why would you want to do this in the first place? BasicUI is the user interface you create to give to the users of your home. Your family members and guests are not going to want to see nor be able to use a bunch of scrolling by log entries. And it will be scrolling by probably just fast enough to make it impossible to read anyway given how little room there is to present any information on BasicUI anyway.
So do you just want this for yourself? Are you using openHABian? If so it’s probably already installed so just select Frontail when you bring up http://openhabian:8080. If not installed, run openhabian-config and install it. If not using openHABian, see https://github.com/mthenw/frontail for instructions for installing and configuring it. Then open Frontail in a separate browser window to watch the logs.
Do you care about specific events? See psyciknz’s reply above to either send an alert or update an Item’s state when that event occurs.
Want to see in a graph when a door was open or closed, see InfluxDB+Grafana persistence and graphing.
You have to explain why you want to do this. What do you want to achieve.
With the information provided, this looks very much like an XY Problem coupled with “when the only tool I have is a hammer all problems look like nails.” Explain what you end goal is and we can probably tell you the best way to achieve it. But it almost certainly will not be by adding a view of your logs scrolling by to BasicUI.