Restore a point between two full backups

Hi,

I make two full backups on Oct 1 and Oct 10. I want to restore the server to a state in Oct 5. So I just do as follows:

1. Perform a transaction log backup on the server on Oct 23. I have never backup transaction log in the past.
2.  Restore the server with Oct 1 full backup with NORECOVERY option.
3. Try to restore to the point at Oct 5 12:00, with the transaction log.

But the restore fails and  SQL Server said the transaction log does not contain the point. The point is too early. Why? Also my .LDF file is about 13G, but the transaction log backup is only 200MB. Why?

Thanks