How to configure log4j to log different log levels to different files for the same logger

What you need to do is have a single <logger> definition with a defined level of INFO, but in your two appender definitions, you set their thresholds accordingly, e.g.

<appender name="ERROR_FILE">
   <param name="Threshold" value="ERROR"/>
</appender>

<appender name="GENERAL">
   <param name="Threshold" value="INFO"/>
</appender>

You then add both appenders to your logger:

<logger name="com.acme">
  <level value="INFO"/>
  <appender-ref ref="ERROR_FILE"/>
  <appender-ref ref="GENERAL"/>
</logger>

Log entries now going to the logger will get sent to both appenders, but since they have different independent thresholds, the ERROR_FILE appender will only log ERROR and above.

Full working solution including the date in the filename:

<appender name="InfoFileAppender" class="org.apache.log4j.rolling.RollingFileAppender">
    <rollingPolicy class="org.apache.log4j.rolling.TimeBasedRollingPolicy">
      <param name="FileNamePattern" value="/var/output/Info_%d{ddMMyyyy}.log" />
    </rollingPolicy>
    <layout class="org.apache.log4j.PatternLayout">
    <param name="ConversionPattern" value="%d{HH:mm:ss} %-5p %t %c{2} - %m%n" />
    </layout>
    <filter class="org.apache.log4j.varia.LevelRangeFilter">
      <param name="levelMin" value="INFO" />
      <param name="levelMax" value="INFO" />
    </filter>
</appender>

<appender name="ErrorFileAppender" class="org.apache.log4j.rolling.RollingFileAppender">
    <param name="Threshold" value="ERROR" />
    <rollingPolicy class="org.apache.log4j.rolling.TimeBasedRollingPolicy">
        <param name="FileNamePattern" value="/var/output/Error_%d{ddMMyyyy}.log" />
    </rollingPolicy>
    <layout class="org.apache.log4j.PatternLayout">
        <param name="ConversionPattern" value="%d{HH:mm:ss} %-5p %t %c{2} - %m%n" />
    </layout>
</appender>
<root>
    <level value="INFO" />
    <appender-ref ref="InfoFileAppender" />
    <appender-ref ref="ErrorFileAppender" />
</root>

You need to use log4j filters:

<filter class="org.apache.log4j.varia.LevelRangeFilter">
    <param name="levelMin" value="ERROR" />
    <param name="levelMax" value="ERROR" />
</filter>

That way you can create log files for each level separately.