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  4. The number of solutions of the equation log 4(x-1)= log 2(x-3) is

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Q. The number of solutions of the equation $\log _{4}(x-1)=\log _{2}(x-3)$ is

JEE MainJEE Main 2021Complex Numbers and Quadratic Equations

Solution:

$\log _{4}(x-1)=\log _{2}(x-3)$
$\Rightarrow \frac{1}{2} \log _{2}(x-1)=\log _{2}(x-3)$
$\Rightarrow \log _{2}(x-1)^{1 / 2}=\log _{2}(x-3)$
$\Rightarrow (x-1)^{1 / 2}=x-3$
$\Rightarrow x-1=x^{2}+9-6 x$
$\Rightarrow x^{2}-7 x+10=0$
$\Rightarrow (x-2)(x-5)=0$
$\Rightarrow x=2,5$
But $x \neq 2$ because it is not satisfying the domain of given equation i.e $\log _{2}( x -3) \to$ its domain $X\,>3$
finally $x$ is $5$
$\therefore $ No. of solutions $=1$