Question

The number of solution(s) of ${\log }_4(x-1)={\log }_2(x-3)$ is/are :

• A. $3$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is B $1$

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

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

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

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

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

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

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

Solutions are $5$ and $2$

But $2$ can't be a solution, since the $\log$ term in LHS will become negative and undefined.

Hence, only one solution exist for the given equation.