Question

The number of solutions of ${\log }_4(x-1)={\log }_2(x-3)$ is

• A. 3
• B. 1
• C. 2
• D. 0

Answer 1

Answer: (B)

${\log }_4(x-1)={\log }_2(x-3)={\log }_{4^{1/2}}(x-3)$
$\Rightarrow {\log }_4(x-1)=2{\log }_4(x-3)$
$\Rightarrow {\log }_4(x-1)={\log }_4(x-3{)}^2$
$\Rightarrow (x-1)=(x-3{)}^2$
$\Rightarrow x^2+9-6x=x-1$
$\Rightarrow x^2-7x+10=0$
$\Rightarrow (x-2)(x-5)=0$
$\Rightarrow x=2$ or $x=5$
Hence, $x=5$
[$\because x=2$ makes $\log (x-3)$ undefined.]