You could simply use the built-in archiving functionality. This setting will keep 7 old log files in addition to your current log. The cleanup is done by NLog automatically.
<?xml version="1.0" ?>
<nlog xmlns="http://www.nlog-project.org/schemas/NLog.xsd"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<targets>
<target name="file" xsi:type="File"
layout="${longdate} ${logger} ${message}"
fileName="${basedir}/logs/logfile.txt"
archiveFileName="${basedir}/logs/log.{#}.txt"
archiveEvery="Day"
archiveNumbering="Rolling"
maxArchiveFiles="7"
concurrentWrites="true" />
</targets>
<rules>
<logger name="*" minlevel="Debug" writeTo="file" />
</rules>
</nlog>
See also the documentation of the file target
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