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} = 2^½
and using Rules for exponents?
2^{2(2x - 1)} = 2^½

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