How can I turn off the Automatic Logging feature? I want to only log private chats and not build a log file of every chat room that I visit.

tools/options -> logging -> automatic log options

You'll want to uncheck 'channel' and 'status'

Also note that query windows and dcc-chat share the same log file, so if you leave the box checked to automatically reload the chat log, it can load prior activity from both.

When I click on Tools and then Options, Logging is not on the options list that appears and, hence, I cannot get to Automatic Logging Options to uncheck the Channel and Status. I am running the 64 bit version of mIRC v7.52 on a Win 10 PC.

i should have said options -> IRC -> logging.

most people keep the options menu expanded so they can see all the options. If you click on the "+" next to "IRC" it expands to show 5 rows, one of them is Logging.

Your windows version will have no effect on this item, and it's been in that same place for many versions across many years.