I found a sample properties file in the Apache Tomcat 7 documentation, but since this is meant for use with Log4j 1.x, I had to adapt it to the Log4j2 properties file syntax. This is the result:
# 'status' refers to log messages from Log4j2 itself
monitorInterval = 30
status = warn
property.loglevel.catalina = info
property.loglevel.localhost = info
property.layoutPattern.catalina = %d %-5p [%t] %-22.22c{1} %m%n
property.layoutPattern.localhost = %d %-5p [%t] %-30.30c{1} %m%n
# Roll-over the logs once per month using CronTriggerPolicy.
property.fileDatePattern.catalina = %d{yyyy-MM}
property.fileDatePattern.localhost = %d{yyyy-MM}
property.cronTriggerSchedule.catalina = 0 0 0 1 * ?
property.cronTriggerSchedule.localhost = 0 0 0 1 * ?
## Appenders
# N.B.: - No need to specify 'appenders = CATALINA, LOCALHOST, CONSOLE'
# since these identifiers do not contain '.' characters.
# - The sub-component identifiers 'policies' and 'cron' are arbitrarily
# chosen; the actual type is specified via the 'type' attribute.
# - 'DirectWriteRolloverStrategy' is used automatically since no 'fileName' specified.
appender.CATALINA.type = RollingFile
appender.CATALINA.name = RollingFile-CATALINA
appender.CATALINA.filePattern = ${sys:catalina.base}/logs/catalina.${fileDatePattern.catalina}.log
appender.CATALINA.layout.type = PatternLayout
appender.CATALINA.layout.pattern = ${layoutPattern.catalina}
appender.CATALINA.policies.type = Policies
appender.CATALINA.policies.cron.type = CronTriggeringPolicy
appender.CATALINA.policies.cron.schedule = ${cronTriggerSchedule.catalina}
appender.CATALINA.policies.cron.evaluateOnStartup = true
appender.CATALINA.filePermissions = rw-r-----
appender.CATALINA.fileOwner = tomcat
appender.CATALINA.fileGroup = adm
appender.LOCALHOST.type = RollingFile
appender.LOCALHOST.name = RollingFile-LOCALHOST
appender.LOCALHOST.filePattern = ${sys:catalina.base}/logs/localhost.${fileDatePattern.localhost}.log
appender.LOCALHOST.layout.type = PatternLayout
appender.LOCALHOST.layout.pattern = ${layoutPattern.localhost}
appender.LOCALHOST.policies.type = Policies
appender.LOCALHOST.policies.cron.type = CronTriggeringPolicy
appender.LOCALHOST.policies.cron.schedule = ${cronTriggerSchedule.localhost}
appender.LOCALHOST.policies.cron.evaluateOnStartup = true
appender.LOCALHOST.filePermissions = rw-r-----
appender.LOCALHOST.fileOwner = tomcat
appender.LOCALHOST.fileGroup = adm
# Uncomment if you want to keep logging to catalina.out after Log4j2 takes over.
#appender.CONSOLE.type = Console
#appender.CONSOLE.name = STDOUT
#appender.CONSOLE.layout.type = PatternLayout
## Configure which loggers log to which appenders
rootLogger.level = ${loglevel.catalina}
rootLogger.appenderRef.CATALINA.ref = RollingFile-CATALINA
#rootLogger.appenderRef.stdout.ref = STDOUT
# Here, the identifier does contain '.' characters, so we must specify the list.
loggers = org.apache.catalina.core.ContainerBase.[Catalina].[localhost]
logger.org.apache.catalina.core.ContainerBase.[Catalina].[localhost].name = LOCALHOST
logger.org.apache.catalina.core.ContainerBase.[Catalina].[localhost].level = ${loglevel.localhost}
logger.org.apache.catalina.core.ContainerBase.[Catalina].[localhost].appenderRef.LOCALHOST.ref = RollingFile-LOCALHOST
The main reason for me to use Log4j2 was to be able get monthly log rotation, but you can easily adjust everything how you like, even without restarting Tomcat.