Not by default, you should create a JavaScript action that retrieves that data, and returns it to the Nanoflow, then you should be able to use that data in the Log Message action
E.g. in the browser console you can type mx.session.getUserAttribute("Name"); to get the name, or use
mx.session.getUserName();
See also https://apidocs.mendix.com/7/client/mx.session.html