The location of the alfresco.log file is set in the webapps/alfresco/WEB-INF/classes/log4j.properties file. In this file you'll see that the alfresco.log file doesn't have a path specified. You might want to change it to something like this and create a logs directory. Otherwise, the file will get created relative to the path of the command that launched tomcat.log4j.appender.File.File=/home/ac/alfresco/logs/alfresco.log
Likewise, you probably want to set the location of the repository files. The proper way to do this is described in the docs (http://wiki.alfresco.com/wiki/Repository_Configuration#New_Installations) but if you just want to get going, edit the webapps/alfresco/WEB-INF/classes/alfresco/repository.properties file and set the dir.root.dir.root=/home/ac/alfresco/alf_data
Make sure those directories are writable by the tomcat user. Also, you do need MySQL running, Tomcat won't start it.