Answer
$x=-\frac{1}{5}$
Work Step by Step
We are given the exponential equation $8^{1-x}=4^{x+2}$.
We can express each side using a common base and then solve for $x$.
$8^{1-x}=(2^{3})^{1-x}=2^{3-3x}$
$4^{x+2}=(2^{2})^{x+2}=2^{2x+4}$
$2^{3-3x}=2^{2x+4}$
Take the natural log of both sides.
$ln(2^{3-3x})=ln(2^{2x+4})$
$(3-3x)ln(2)=(2x+4)ln(2)$
Divide both sides by $ln(2)$.
$3-3x=2x+4$
Add $3x$ to both sides.
$3=5x+4$
Subtract 4 from both sides.
$5x=-1$
Divide both sides by 5.
$x=-\frac{1}{5}$
