Answer
$x=-\frac{7}{2}$
Work Step by Step
We are given that $4^{3x-7}=32^{2x}$. Both of these numbers are powers of 2, so we can rewrite the equation as $(2^{2})^{3x-7}=(2^{5})^{2x}$.
Therefore,
$2^{6x-14}=2^{10x}$
From the uniqueness of $b^{x}$, we know that $b^{x}=b^{y}$ is equivalent to $x=y$ (when $b\gt0$ and $b\ne1$).
Therefore, $6x-14=10x$.
Subtract 6x from both sides.
$4x=-14$
Divide both sides by 4.
$x=-\frac{14}{4}=-\frac{7}{2}$
