The Batman logo that you can see below have been used until 1986. In that period some artists had fun doing odd additions to the bat shape, or distorting or shattering the logo for particular covers, and occasionally the bat shape was dropped altogether, as on issue 400, below.
At any rate the Year One issues sold like hotcakes, and the Batman editorial staff realized they were on to something. They quickly started developing new short story arcs, hoping to capture the same audience. The first one to come out was this one:
This time the logo and story title were combined into a rectangle, and the cover art was designed around it, a departure from the usual process, but since it was only for four issues, not hard to do. A logo area of this size and shape is rather an awkward fit for a comics cover, but a good artist, in this case Mike Zeck, can make the dead areas at the top work for him with a little planning.
For the first time the word BATMAN has been created with type, specifically the font Belwe Bold here. It’s artfully composed by DC staffer Keith “Kez” Wilson, in charge of cover design at the time. I believe he also created the story title and the whole logo box. The splattery word BEAST gives good contrast to the staid BATMAN, and I think it works quite well. Belwe is what’s called a slab-serif font, meaning that the little caps at the ends of the letter strokes (like at the upper right corner of the N) are squared-off slabs, rather than coming to a point as in many other serif fonts. The use of a larger beginning and ending letter, with a small line below the others, is a pleasing design.