Agent output to file cannot be done through Sync Center. You will need to synchronize on-demand from the command line in order to enable verbose agent logging to file with SQL Server Express. To run the Merge Agent from the command line, execute replmerg.exe from the COM folder. It will look something like this:
This example outputs the verbose Merge Agent log to C:\TEMP\mergeagent.log. You will need to update the replmerg.exe parameter values according to your environment.