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  4. The sum of the roots of the equation x+1-2 log 2(3+2x)+2 log 4(10-2-x)=0, is :

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Q. The sum of the roots of the equation $x+1-2 \log _{2}\left(3+2^{x}\right)+2 \log _{4}\left(10-2^{-x}\right)=0$, is :

JEE MainJEE Main 2021Complex Numbers and Quadratic Equations

Solution:

$x+1-2 \log _{2}\left(3+2^{x}\right)+2 \log _{4}\left(10-2^{-x}\right)=0 $
$\log _{2}\left(2^{x+1}\right)-\log _{2}\left(3+2^{x}\right)^{2}+\log _{2}\left(10-2^{-x}\right)=0$
$\log _{2}\left(\frac{2^{x+1} \cdot\left(10-2^{-x}\right)}{\left(3+2^{x}\right)^{2}}\right)=0$
$\frac{2\left(10.2^{x}-1\right)}{\left(3+2^{x}\right)^{2}}=1 $
$\Rightarrow 20.2^{x}-2=9+2^{2 x}+6.2^{x} $
$\therefore \left(2^{x}\right)^{2}-14\left(2^{x}\right)+11=0$
Roots are $2^{x_{1}} \& 2^{x_{2}}$
$\therefore 2^{ x _{1}} \cdot 2^{ x _{2}}=11$
$ x _{1}+ x _{2}=\log _{2}(11)$