Question

Sum of all possible values of $x$ which satisfy the equation ${\left(\log \right)}_3(x-3)={\left(\log \right)}_9(x-1)$ is:

• A. $2$
• B. $5$
• C. $7$
• D. $10$

Answer 1

Answer: (B)

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