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4-2.Quadratic Equations and Inequations
hard
The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is :
A
$1+\log _6(8)$
B
$\log _8(6)$
C
$1+\log _8(6)$
D
$\log _8(4)$
(JEE MAIN-2024)
Solution
$ (8)^{2 x}-16 \cdot(8)^x+48=0 $
$ \text { Put } 8^x=t $
$ t^2-16+48=0 $
$ \Rightarrow t=4 \text { or } t=12 $
$ \Rightarrow 8^x=4 \quad 8^x=12 $
$ \Rightarrow x=\log _8 x \quad x=\log _8 12 $
$ \text { sum of solution }=\log _8 4+\log _8 12 $
$ =\log _8 48=\log _8(6.8) $
$ =1+\log _8 6$
Standard 11
Mathematics
