Question

If ${\log }_2\left(9^{x-1}+7\right)-{\log }_2\left(3^{x-1}+1\right)=2$, then the sum of value(s) of $x$ is equal to

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

${\log }_2\left(\frac{9^{x-1}+7}{3^{x-1}+1}\right)=2$
$\frac{{\left(3^{x-1}\right)}^2+7}{3^{x-1}+1}=4\text{ Put }{3}^{x-1}=t$
$t^2+7=4t+4\Rightarrow t^2-4t+3=0\Rightarrow (t-1)(t-3)=0$
$t=1\Rightarrow 3^{x-1}=1\Rightarrow x-1=0\Rightarrow x=1$
$t=3\Rightarrow 3^{x-1}=3\Rightarrow x-1=1\Rightarrow x=2$