UnauthorizedAccessException in RollingFileAppender when acquiring a mutex

I have a setup where I want two processes, running as different user accounts to log to the same file.

I have set up the file permissions so they work, but the second process that starts cannot perform the logging due to this error:

log4net:ERROR Could not create Appender [myAppender] of type [log4net.Appender.RollingFileAppender]. Reported error follows.
System.UnauthorizedAccessException: Access to the path 'E__Log_MyLog.log' is denied.
at System.IO.__Error.WinIOError(Int32 errorCode, String maybeFullPath)
at System.Threading.Mutex.MutexTryCodeHelper.MutexTryCode(Object userData)
at System.Runtime.CompilerServices.RuntimeHelpers.ExecuteCodeWithGuaranteedCleanup(TryCode code, CleanupCode backoutCode, Object userData)
at System.Threading.Mutex.CreateMutexWithGuaranteedCleanup(Boolean initiallyOwned, String name, Boolean& createdNew, SECURITY_ATTRIBUTES secAttrs)
at System.Threading.Mutex..ctor(Boolean initiallyOwned, String name, Boolean& createdNew, MutexSecurity mutexSecurity)
at System.Threading.Mutex..ctor(Boolean initiallyOwned, String name)
at log4net.Appender.RollingFileAppender.ActivateOptions()
at log4net.Repository.Hierarchy.XmlHierarchyConfigurator.ParseAppender(XmlElement appenderElement)

I think the mutex should be created with access for Everyone, and that would solve the problem.