Answer
$x=-\frac{1}{4}$
Work Step by Step
We are given the exponential equation $4^{x}=\frac{1}{\sqrt2}$.
We can express each side using a common base and then solve for $x$.
$4^{x}=(2^{2})^{x}=2^{2x}$
$\frac{1}{\sqrt2}=\frac{1}{2^{\frac{1}{2}}}=2^{-\frac{1}{2}}$
$2^{2x}=2^{-\frac{1}{2}}$
Take the natural log of both sides.
$ln(2^{-\frac{1}{2}})=ln(2^{-\frac{1}{2}})$
$2xln(2)=-\frac{1}{2}ln(2)$
Divide both sides by $ln(2)$.
$2x=-\frac{1}{2}$
Divide both sides by 2.
$x=-\frac{1}{4}$
