Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

${\log }_4(x-1)={\log }_2(x-3)$

$\Rightarrow {\log }_{2^2}(x-1)={\log }_2(x-3)$

$\Rightarrow \frac{1}{2}{\log }_2(x-1)={\log }_2(x-3)$

$\Rightarrow {\log }_2(x-1)=2{\log }_2(x-3)$

$\Rightarrow (x-1)={(x-3)}^2$

$\Rightarrow x-1=x^2-6x+9$

$\Rightarrow x^2-6x-x+9+1=0$

$\Rightarrow x^2-7x+10=0$

$\Rightarrow x^2-2x-5x+10=0$

$\Rightarrow x(x-2)-5(x-2)=0$

$\Rightarrow (x-2)(x-5)=0$

$\therefore x=2,5$

Number of solutions$=2$