Yes, it is very much possible.
To log your conversations:
1) Open mIRC
2) Press ALT + O or go to the bar on top and click the icon with a hammer and folder.
3) On the left, go to the IRC +'d box and turn it into a - .
4) Click on Logging.
5) Go to the drop box at the top. (Assuming you are using mIRC 6.14) Click whether you want to log private messages, channels, or both.
6) Select any other options you may want, such as time stamping.
7) Click OK at the bottom.
Hope this helps.
aka "The Big 'Z'"