How to delete records without a primary key in a stardard way (i.e. not using ctid)
CAUTION : This can make you lose data, so consider using transaction block. 

I think this is the more generic solution: 

If you consider that the line you want to delete is:

select * from the_log limit 1 


then if you have one column that has a unique value... 

delete from the_log  
where unique_the_log_column in 
(select unique_log_column from the_log limit 1); 


If you do not have a unique column, but you have two columns that combined generates a unique value:  

delete from the_log  
where (col1, col2) in (select col1, col2 from the_log limit 1); 


Where any_the_log_column is a column with a unique value in this table. 

This will probably delete the first line of the log table in any dbms - if data in column is not repeated, but remember that tables sometimes is not selected ordered in some databases so you must have a id in this table... 

Consider creating a id column and increasing it by the numbers of the lines in the table, based on date or something if you don't want to get hurt in the future...