Why can a log backup not be taken without first taking a full backup?
When a database is switched to FULL recovery, it behaves as if it was in SIMPLE recovery until you take a full database backup. Up to that moment, you cannot take log backups, because the log gets truncated on CHECKPOINT, as happens in SIMPLE recovery.

See this article by Paul Randal for reference.

Log backups form part of a chain that allow the whole database to be recovered. The chain starts with a full backup (complete copy), and may continue from that point with further differential or log backups.



Image taken from Complete Database Restores (Full Recovery Model)

A transaction log backup contains the information needed to:


Redo all committed changes to the database since the last full or log backup
Undo in-progress (not yet committed) changes at the time the log backup was taken


In principle, it would be possible to recreate a database from log information alone, if every log backup ever was available, including the initial creation of the database itself, but that is not the way it was implemented.

As it is, a full backup is necessary is start the recovery path.