Environment
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CloudBees CI (CloudBees Core) on modern cloud platforms - Managed controller
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CloudBees CI (CloudBees Core) on modern cloud platforms - Operations Center
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CloudBees CI (CloudBees Core) on traditional platforms - Client controller
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CloudBees CI (CloudBees Core) on traditional platforms - Operations Center
 
Explanation
Jenkins periodic tasks log are stored under the log root directory that defaults to $JENKINS_HOME/logs. The files are rotated once a day and up to 6 files are kept. But files are not limited in size. At least until JENKINS-64151: Adjust file size rotation for AsyncPeriodicWork / AsyncAperiodicWork Task Logs is resolved.
| If a specific task is generating a lot of logs, it is usually indicative of a problem with that particular task and an issue must be logged. | 
Resolution
There are ways to control the rotation via system properties (see How to add Java arguments to Jenkins and avoid disk space issues if a specific task generates too much log.
Change the Time rotation of all tasks
To change the default time rotation in minutes of all tasks - that is 1440 minutes (1 day) by default) - set the system property hudson.model.AsyncPeriodicWork.logRotateMinutes and hudson.model.AsyncAperiodicWork.logRotateMinutes on startup. For example, change the default file time rotation of all tasks to 12 hours: -Dhudson.model.AsyncPeriodicWork.logRotateMinutes=720 and -Dhudson.model.AsyncAperiodicWork.logRotateMinutes=720.
Change the Time rotation of a specific tasks
To change the time rotation in minutes of a specific task - that is 1440 minutes (1 day) by default - set the system property <FullyQualifiedClassName>.logRotateMinutes on startup. For example, change the hudson.model.WorkspaceCleanupThread file time rotation to 12 hours: -Dhudson.model.WorkspaceCleanupThread.logRotateMinutes=720.
Change the Size rotation of all tasks
To change the default size rotation in bytes of all tasks - that is -1 (unlimited) by default - set the system property hudson.model.AsyncPeriodicWork.logRotateSize and hudson.model.AsyncAperiodicWork.logRotateSize on startup. For example, change the default file size rotation of all tasks to 10 MB: -Dhudson.model.AsyncPeriodicWork.logRotateSize=10485760 and -Dhudson.model.AsyncAperiodicWork.logRotateSize=10485760.
Change the Size rotation of a specific tasks
To change the size rotation in bytes of a specific task - that is -1 (unlimited) by default - set the system property <FullyQualifiedClassName>.logRotateSize on startup. For example, change the hudson.model.WorkspaceCleanupThread file size rotation to 10 MB: -Dhudson.model.WorkspaceCleanupThread.logRotateSize=10485760.
Change the default location of all tasks
To change the default log root location in Jenkins - that is $JENKINS_HOME/logs by default - set the system property hudson.triggers.SafeTimerTask.logsTargetDir.