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 Main · 2021 · 31 Aug 2021 · Shift 2 · Basic Mathematics

The sum of the roots of the equation,

x + 1 - 2 lo g_{2} (3 + 2^{x}) + 2 lo g_{4} (10 - 2^{- x}) = 0

, is:

Answer: (B)

Put

t = 2^{x}

, so

x = lo g_{2} t

and

2 lo g_{4} (10 - 2^{- x}) = lo g_{2} (10 - \frac{1}{t})

. The equation becomes

lo g_{2} \frac{2 t ( 10 - 1/ t )}{( 3 + t ) ^{2}} = 0

, i.e.

20 t - 2 = (3 + t)^{2}

, giving

t^{2} - 14 t + 11 = 0

. Its roots are positive with product

11

, and each gives a root

x = lo g_{2} t

of the original equation. The sum of the

x

-roots is

lo g_{2} t_{1} + lo g_{2} t_{2} = lo g_{2} (t_{1} t_{2}) = lo g_{2} 11

JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Single correct option