Question

If the fourth term in the binomial expansion of $\bigg(\frac{2}{x} + x^{log_8 x}\bigg)^6 (x > 0) \, is \, 20 \times 8^7$, then a value of $x$ is :

• A. $8$
• B. $8^{2}$
• C. $8^{-2}$
• D. $8^{3}$

Answer 1

Answer: (B)

$T_4 \, = \, T_{3+1} = \bigg(^{6}_{3}\bigg) \bigg(\frac{2}{x}\bigg)^3 . \bigg(x^{log_g x}\bigg)^3$
$20 \times 8^7 = \frac{160}{x^3}. x^{3 log_g x}$
$8^6 = x^{log_2 x }-3$
$2^{18} \, = \, x^{log_2 x-3}$
$\Rightarrow \, \, 18 \, = \, (log_2 x- 3) (log_2 x)$
Let $log_2 $ x = t
$\Rightarrow \, \, t^2 - 3t - 18 = 0$
$\Rightarrow $ (t - 6)(t + 3) = 0
$\Rightarrow $ t = 6, - 3
$log_2 x = 6 \Rightarrow x = 2^6 = 8^2$
$log_2 x = -3 \Rightarrow x = 2^{-3} = 8^{-1}$