Answer
$\displaystyle \{\frac{\log 7}{\log 3-\log 5+\log 7} \}$ or $\{1.356\}$
Work Step by Step
... Apply $\log$(...) to both sides and isolate x
$ \begin{aligned}
\left(\displaystyle \frac{3}{5}\right)^{x}&=7^{1-x}\\
\log\left(\displaystyle \frac{3}{5}\right)^{x}&=\log\left(7^{1-x}\right)\\
x\log(\displaystyle \frac{3}{5})&=(1-x)\log7\\
x\log(\displaystyle \frac{3}{5})&=\log7-x\log7\\
x\log(\displaystyle \frac{3}{5})+x\log7&=\log7\\
x(\log3-\log5+\log7)&=\log7
\end{aligned}$
$x=\displaystyle \frac{\log 7}{\log 3-\log 5+\log 7}\approx 1.356$
Solution set: $\displaystyle \{\frac{\log 7}{\log 3-\log 5+\log 7} \}$ or $\{1.356\}$
