Linux Syslog – Remove ‘Last Message Repeated x Times’ from Logs

Logrotate can be made to only keep one copy of a logfile... If you RTFM you'll find the following bit regarding configuration settings:

rotate count
    Log files are rotated count times before being removed or mailed to the 
    address specified in a mail directive. If count is  0,  old  versions  
    are  removed  rather than rotated.

You can couple rotate with size, again from the logrotate(8) man page, to keep the file size small. While not by number of lines but by k, M, G size.

size size
    Log  files  are rotated when they grow bigger than size bytes. If size is 
    followed by M, the size if assumed to be in megabytes.  If the G suffix is 
    used, the size  is  in gigabytes.   If  the k is used, the size is in 
    kilobytes. So size 100, size 100k, and size 100M are all valid.

I'm not sure I get the connection between VNC and MySQL here, but if you want to stop something logging to a file, make the log file a soft link to /dev/null. If a process is logging to syslog, you can either edit the syslog config file to make it stop logging that process, or simply kill syslog outright.

Logs are good for figuring out what problems you are having on the system. Usually high log load, on a non-debug enabled system, is an indication of something going horribly wrong.