Answer
$x=\frac{1}{2}$
Work Step by Step
We rewrite both sides, so that the base is $e$ on both sides.
$e^{4x-1}=(e^2)^x$
By the law of exponents: $(a^x)^y=a^{xy}$
$(e^2)^x=e^{2x}$
Thus,
$e^{4x-1} = e^{2x}$
Both sides have the same base, so set the exponents equal to have:
$4x-1=2x$
$4x=2x+1$
$4x-2x=1$
$2x=1$
$x=\frac{1}{2}$
