transaction log after restore
A full database backup contains almost no transactions - just the transactions that happened while the FULL transaction log was being created.
The LDF file created when a database is restored using a full database backup is almost empty, regardless its size. It should be inflated (with no data but active portion of transaction log) to its backed up size when restoring.

If you want to restore the transaction log file too, the best way to do that is to restore the database using its LDF and LDF files (attach them), instead of using a full database backup.

There are many reasons why the original file is huge:


The database is in FULL recovery model and you don't create transaction log backups at all
The database is in FULL recovery model and you don't create transaction log backups often enough
there are large-scale index operations 


For any of the possible causes, creating transaction log backups regularly should keep the size of the LDF file.