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)

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