It's needed for a couple of small things like transaction rollbacks and crash recovery on restart (ensuring a consistent database)
If there was no transaction log, then any failure in any data modification, any explicit rollback, any uncontrolled shutdown would require that the database be restored from backup afterwards as there would be no other way to get back to a transactionally consistent state. Except that full backups need the transaction log to be transactionally consistent too.
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: Discussions on DB performance with occasional diversions into recoverability
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