Question

The integral root of the equation ${\log }_{\frac{2+x}{10}}7={\log }_{\frac{2}{x+1}}7$ is

Answer 1

The given equation is equivalent to
$\{\begin{array}{c}\frac{2}{x+1}>0,\ne 1\\ \frac{2+x}{10}>0,\ne 1\\ \frac{2+x}{10}=\frac{2}{x+1}\end{array}$
$\Rightarrow \{\begin{array}{c}x+1>0\\ x\ne 1,8\\ (2+x)(x+1)=20\Rightarrow x=-6,3\end{array}$
$\therefore x=3$ is root of the original equation.