Answer
$x=\frac{9}{5}$
Work Step by Step
If $b\gt0$ and $b\ne1$, then $y=log_{b}x$ means $x=b^{y}$ for every $x\gt0$ and every real number $y$.
Therefore, $log_{7}(5x-2)=1$ is equivalent to $7^{1}=7=5x-2$.
For $7=5x-2$, add 2 to both sides.
$5x=9$
Divide both sides by 5.
$x=\frac{9}{5}$
