Question

If ${\log }_{\frac{2+x}{10}}7={\log }_{\frac{2}{x+1}}7,$ then the value of $x$ is/are

• A. $-6$
• B. $3$
• C. $6$
• D. $-3$

Answer 1

The correct option is B $3$

${\log }_{\frac{2+x}{10}}7={\log }_{\frac{2}{x+1}}7$ is defined if
$\frac{2+x}{10}>0,\frac{2+x}{10}\ne 1\Rightarrow x>-2,x\ne 8...\left(1\right)\frac{2}{x+1}>0,\frac{2}{x+1}\ne 1\Rightarrow x>-1,x\ne 1...\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$
$x>-1,x\ne 1,8...\left(3\right)$

${\log }_{\frac{2+x}{10}}7={\log }_{\frac{2}{x+1}}7\Rightarrow \frac{2+x}{10}=\frac{2}{x+1}\Rightarrow x^2+3x-18=0\Rightarrow (x+6)(x-3)=0\Rightarrow x=3(\because x>-1)$