If the given function $f(x)=\frac{\log (7+x)}{\log (3+x)}, x>\,0$ is
decreasing function then $f^{\prime}(x)<0$ $\Rightarrow \frac{\log (3+x) \frac{1}{(7+x)}-\frac{1}{(3+x)} \log (7+x)}{[\log (3+x)]^{2}}<\,0$
$\Rightarrow \, \frac{\log (3+x) \frac{1}{(7+x)}-\frac{1}{(3+x)} \log (7+x)}{[\log (3+x)]^{2}}<\,0$
$\Rightarrow \,(3+x)^{(3+x)}<(7+x)^{(7+x)}$ , it is true for every value of $ x>\,0$
$\Rightarrow \, x \in(0, \infty)$