When the system log file increases it means that there is some issue associated software or applications. In that case simply check the latest activities in the system log file.
For that login to your server and execute the following command:
If the issue is not encountered from the latest activities, you can go through the complete log file for hints:
If the same issue is repeating, search for the repeated messages in the log file. Most of the time, its an error message, but can also be a warning to disable something by which its causing. Sometimes, it may be an intrusion attack, which you can check in to the security log file.
Once the problem is encountered, fix it and clear the log file and start with an empty one. Continuing the same large log file will consume much space. Enter the following command:
logrotate -v -f /etc/logrotate.d/logname
The above command will archive the old log file, and creates a new empty log file for further use.