Question

The number of solutions of $\log_{2}\left(x-1\right)=2\,\log_{2}\left(x-3\right)$ is

• A. 2
• B. 1
• C. 6
• D. 7

Answer 1

Answer: (B)

Given, $\log _{2}(x-1)=2 \log _{2}(x-3)$
$\Rightarrow \log _{2}(x-1)=\log _{2}(x-3)^{2}$
$\Rightarrow x-1=(x-3)^{2}$
$\Rightarrow x-1=x^{2}+9-6 x$
$\Rightarrow x^{2}-7 x+10=0$
$\Rightarrow (x-2)(x-5)=0$
$\Rightarrow x=2,5$
Since, $x=2$ does not satisfis the equation.
So, $x=5$ is the only solution.
Hence, number of solution is one.