The sum of all the solutions of the equation ( 8 ) 2 x - 16 · ( 8 ) x + 48 = 0 is : [2024]

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Solution

Q.
The sum of all the solutions of the equation ${(8)}^{2 x} - 16 \cdot {(8)}^{x} + 48 = 0$ is : [2024]

1 $1 + \log_{8} (8)$

2 $\log_{8} (6)$

3 $1 + \log_{8} (6)$

4 $\log_{8} (4)$

Ans.
(3)

We have, $8^{2 x} - 16 \cdot 8^{x} + 48 = 0$

Let $8^{x} = t$

$\Rightarrow t^{2} - 16 t + 48 = 0 \Rightarrow t^{2} - 12 t - 4 t + 48 = 0$

$\Rightarrow (t - 12) (t - 4) = 0 \Rightarrow t = 4 or t = 12$

$\Rightarrow 8^{x} = 4 or 8^{x} = 12$

$\Rightarrow x = \log_{8} 4 or x = \log_{8} 12$

Sum of solutions $= \log_{8} 4 + \log_{8} 12 = \log_{8} 48 = \log_{8} (6 \cdot 8)$

$= \log_{8} 8 + \log_{8} 6 = 1 + \log_{8} 6$

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