Starting equation: 42x - 1 =

 2

Re-write 4 = 22, and re-write

 2
as 2½

The above two revisions result in this equation:
[22]2x - 1 = 2½
and using Rules for exponents?
22(2x - 1) = 2½

If the powers are equal, and the bases (here, 2) are equal then the exponents must also be equal:

2(2x - 1) = ½
4x - 2 =½
4x = 2.5
x = 0.625