Starting equation: 4^{2x  1} = 

Rewrite 4 = 2^{2}, and rewrite 

as 2^{½} 
The above two revisions result in this equation:
[2^{2}]^{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 