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, ${\log }_2(x-1)=2{\log }_2(x-3)$
$\Rightarrow {\log }_2(x-1)={\log }_2(x-3{)}^2$
$\Rightarrow x-1=(x-3{)}^2$
$\Rightarrow x-1=x^2+9-6x$
$\Rightarrow x^2-7x+10=0$
$\Rightarrow (x-2)(x-5)=0$
$\Rightarrow x=2,5$
Since, $x=2$ does not satisfis the equation.
So, $x=5$ is the only solution.
Hence, number of solution is one.