I have an Asp.Net MVC 5 website with EntityFramework codefirst approach in a shared hosting plan. It uses the open source WebbsitePanel for control panel and its SQL Server pane
Call your hosting company and either have them set up regular log backups or set the recovery model to simple. I'm sure you know what informs the choice, but I'll be explicit anyway. Set the recovery model to full if you need the ability to restore to an arbitrary point in time. Either way the database is misconfigured as is.