The way I would do it is to start with a rectangle instead of a square center for the log cabin block. That way it would more evenly take up the quilt back. I wouldn't make the block as big as the backing though. I'd put an outside border of one color fabric to frame the log cabin block to make it the size I wanted.
I've drawn an example on graph paper to illustrate what I mean. My imaginary quilt top measures 84" x 92." I want an extra four inches on each side so my quilt back should measure 92" x 100."
I've started my log cabin block center with a 8" x 12" rectangle. I keep adding logs on the sides that measure 8" wide. The logs on the top and bottom measure 12" high. My finished log cabin block measures 64" x 84." I add side border fabric that is 14" wide and top/bottom border fabric that is 8" high. That would make the backing the 92" x 100" size that I wanted.