 
Log4net
Use ${APPDATA}
 <appender name="RollingFileAppender" type="log4net.Appender.RollingFileAppender">
      <file type="log4net.Util.PatternString" value="${APPDATA}\AppName\log" />
      <appendToFile value="true" />
      <rollingStyle value="Date" />
      <datePattern value="_yyyy.MM.dd.TXT" />
      <staticLogFileName value="false" />
      <layout type="log4net.Layout.PatternLayout">
        <conversionPattern value="%date %-5level %logger - %message%newline" />
      </layout>
    </appender>
NLog
Use ${specialfolder:folder=ApplicationData}