If you cannot set up a cron job to run Magento's cron trigger (either cron.sh or cron.php) regularly, then Magento's internal cron jobs don't run.
If your hosting provider won't let you run cron jobs, then you will have to call cron.php periodically by calling it remotely with curl or wget.
Two things to finish out the setup:
you must run
cron.php at least every 15 minutes minimum so you don't get "too late for the schedule" because the job got set but didn't run within the expiry time period. Magento cron jobs are a two step process, the job gets set on the first trigger (scheduled), then then on next trigger, it gets run (at about the scheduled time after the nearest trigger runtime)
you must go to System->Config->Advanced->System to complete the Log Cleaning setup.
Generate schedule information (System->Config->Advanced->System) under the Cron windowshade for a system that's running
cron.sh/cron.php every 15 minutes.
Note: AOE_Scheduler is being run on this system and the extra fields are there for it. If you're running AOE_Scheduler and you're running a cron trigger (
cron.php) every 15 minutes, you need to set the AOE_Scheduler Heartbeat Task Schedule appropriately to be every 15 minutes.