Question

Sum of all possible values of $x$ which satisfy the equation $\left(\log\right)_{3}\left(\right.x-3\left.\right)=\left(\log\right)_{9}\left(\right.x-1\left.\right)$ is:

• A. $2$
• B. $5$
• C. $7$
• D. $10$

Answer 1

Answer: (B)

We have,
$\log _3(x-3)=\log _9(x-1) $
$\Rightarrow \frac{\log x-3}{\log 3}=\frac{\log x-1}{\log 9} $
$\Rightarrow \frac{\log x-3}{\log 3}=\frac{\log x-1}{\log 3^2}$
$\Rightarrow \frac{\log x-3}{\log 3}=\frac{\log x-1}{2 \log 3}$
$\Rightarrow 2 \log x-3=\log x-1 $
$\Rightarrow \log x-3^2=\log x-1 $
$\Rightarrow(x-3)^2=x-1 $
$\Rightarrow x^2-7 x+10=0 $
$\Rightarrow x=2, 5$
$x =2$ is not possible as $\log _3(x-3)$ is not defined for $x=2$.
Therefore, $x=5$.