Answer
$x=\displaystyle \frac{7}{9}$
Work Step by Step
First, the solutions must satisfy $\left\{\begin{array}{l}
x\gt 0\\
x\gt -7
\end{array}\right.\quad \Rightarrow x\gt 0\qquad (*)$
in order for the equation to be defined.
On the LHS, we apply $\quad \displaystyle \log_{a}\frac{M}{N}=\log_{a}M-\log_{a}N$
On the RHS, apply$\quad \log 10^{-1}=-1$
$\displaystyle \log(\frac{x}{x+7})=\log\frac{1}{10}$
... apply the principle of logarithmic equality
$\displaystyle \frac{x}{x+7}=\frac{1}{10}\qquad $... multiply with $10(x+7)$
$10x=x+7$
$9x=7$
$x=\displaystyle \frac{7}{9}$, which satisfies (*), and is a valid solution.
