I'm studying now how floating number is implemented in modern computers. https://en.wikipedia.org/wiki/Single-precision_floating-point_format
I understand that a floating number is represented via three components: sign, exponent, and fraction.
The sign part is super easy since that represents whether the number is positive or negative using one bit. If 8 bits are assigned to the exponent part, why the interpretation would be 2^{x - 127} rather than simple 2^{x}?
It's the fastest way to compare both positive and negative exponents.
You have more information here.