There’s been a lot of questions on the 4GB database size limit on SQL Server Express Edition.
So to clarify, the 4GB (4096MB to be exact) limit is a per database limit. Meaning you can have multiple databases on Express with each one capped at 4GB.
Also, the 4GB limit applies to the data files only (mdf/ndf) and the log files (ldf) does not count in the 4GB limit. So you can have up to 4Gb total for your data files (mdf/ndf) and no cap (unrestricted) size on your log files (ldf).
Hope this helps.