As a rule, I log every CLI session I ever run. It has saved my neck on countless occasions and just this morning saved me a couple of hours of effort tallying serial numbers. It's surprising how often I use these logs to find some forgotten bit of info or to prove that "it really was all working when I left it"!
Every saved session I create in PuTTY is configured to log into a common directory using a file name in the format year-month-day-time-hostname.log, which gives me a nice, searchable, date-sorted record of what I typed and saw for every session I've ever opened.
The best way to do this is by just configuring Session -> Logging to "All session output" and the file name to something appropriate (the string I use is "&Y-&M-&D-&T-&H.log"). If you like, while you're there, go into Window and set the scrollback buffer to 2 million lines. Finally, return to the Session screen and save as "Default Settings".
From then on all your sessions you create, including ones where you just enter the IP and connect, will be logged. Note that any pre-existing saved sessions will need to be edited if they were not originally set to log.
Do it now, you know it makes sense!