ZIP archives supported with NLog 4.0, see news post
Use enableArchiveFileCompressionfor example
<target name="file" xsi:type="File"
layout="${longdate} ${logger} ${message}"
fileName="${basedir}/logs/logfile.txt"
archiveFileName="${basedir}/archives/log.{#}.txt"
archiveEvery="Day"
archiveNumbering="Rolling"
maxArchiveFiles="7"
enableArchiveFileCompression="true" />
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