gives you the space that's used by the diretories and files in them including all subdirectories.
To simplify the output you could do
du -hsx --max-depth=1
which would give you only the totals of all subdirectories.
Maybe you want to play around with the
switch as well.
In you example 'du' will report the total for '/' including all subdirectories. If you don't want that you need to adjust the parameters of 'du'.
Why shouldn't it be possible to have 600GB worth of logs?
Do you run logrotate or any other housekeeping tools?
The detail you missed out in your post is how you partitions are set up. it looks like you have a 20GB partition for root and a 1,8TB partition for /home. What other partitions do you have set up?
To answer that question use 'fdisk -lu' and 'lvdisplay'/'vgdisplay' if applicable.
btw: Which flavour of Linux are you running?