Answer
$x=5$
Work Step by Step
We are given the exponential equation $7^{\frac{x-2}{6}}=\sqrt7$.
We can express each side using a common base and then solve for $x$.
$7^{\frac{x-2}{6}}=7^{\frac{1}{2}}=\sqrt 7$
Take the natural log of both sides.
$ln(7^{\frac{x-2}{6}})=ln(7^{\frac{1}{2}})$
$(\frac{x-2}{6})ln(7)=\frac{1}{2}ln(7)$
Divide both sides by $ln(7)$.
$\frac{x-2}{6}=\frac{1}{2}$
Multiply both sides by 6.
$x-2=3$
Add 2 to both sides.
$x=5$
