Matthew,
Yes, you'd have to increase the global log retention setting (please keep in mind that this is different than 'upping the logging', which usually means increasing the verbosity or defails of the logs).
If you go into the host properties of your master server, on the Global Attributes screen, you will find the 'Keep Logs' field in the 'Cleanup' section of the screen. Changing that, however will keep ALL of your logs for 1 year (unless you manually clean them out).
Actually, that might be a better way for you to do this than copying off the log files periodically as we've suggested. I suspect that if you copy off the logs and then copy them back at some point, they would be usable with the status of backup report, but I'm not 100% sure of that. If, on the other hand, you set the log retention to 1 year and then manually cleaned out all the *other* logs, then I'm very sure you'll achieve what you're after.
Of course, all this depends on how much disk space you have on the master server and how big each of your log files are ... You may need to tune your system accordingly.
OK, 1 quick update (did some poking around): I don't think the 'Status of Backups' Report is actually tied to the presense of log files; I think it's stored in the jobs db or something. Why do I think this? Because I can pull up that report for clients over the last couple years on my server and I *know* that we clean out the log files weekly.
DP
PS. Matthew, please don't forget to assign some points when you feel that your question has been addressed meaningfully. Thanks!