The logs are saved in mIRC's Compatibility folder.
If you go to C:\Program Files\mIRC you'll see a button on the top called "Compatibility Files". The only reason it does this is because mIRC is running "virtualized" so any writes to Program Files are redirected to C:\Users\Username\AppData\VirtualStore\ProgramFiles\mIRC.
mIRC will need to run as Administrator or UAC will need to be shut off in order to have it write directly to the folder.