Q: memory problems

Suffice it to say, when you had 20 GB free, that's when you should have started being concerned.  If this happened without anything prior to it, your Mac may just need to run overnight with Energy Saver turned off, but screen saver turned on.

That would clean up the excess log files.

If your data is backed up, you can try forcing the situation by running Applications -> Utilities ->Terminal to do

sudo update daily

sudo update weekly

sudo update monthly

With your administrator password.  This forces the log files to clean up.  Do so only with the machine completely disconnected from the internet.

Restart the computer when done to return it to non-sudo mode.  The Mac will remain in "superuser" mode for a specific time period, where it could be vulnerable until it automatically logs off.    This is why I say restart when done, as that forces it out of the sudo mode.

If you still have trouble cleaning it, the end of my tip:

http://discussions.apple.com/docs/DOC-1689

should help.

Also you may want to look at my FAQ*:

http://www.macmaps.com/diskfull.html

* Links to my pages may give me compensation.

Disk Space - Free Up

Disk Space – Free Up (2)

Disk Space - Free Up - Mavericks

Disk Space Filling Up – OmniDiskSweeper

Disk Space Filling Up - WhatSize

Drive space – minimum free required

Disk Space - Other