Recording a bash session
Solution 1:
You can use script to log all outputs from a command, or even to start a command. One command which you could start is a new issue of a shell. E.g.
script my_log /usr/local/bin/bash
If this is not a once off then please mention that in your post. (In which case you probably want to set it in the login scripts for that user, possibly exec script and placing the log somwhere where the log cannot be manipulated).