GATE Civil 2023 Set 2 | GA Question: 9

Question

If $x$ satisfies the equation $4^{8^{x}}=256$, then $x$ is equal to __________.

• A. $\frac{1}{2}$ 
• B. $\log _{16} 8$
• C. $\frac{2}{3}$
• D. $\log _{4} 8$ 

Answer 1

Given that, $4^{8^{x}}=256$

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

$\Rightarrow 8^{x}=4 \quad (\because \text{In the same base, we can compare the powers})$

$\Rightarrow (2^3)^x = 2^2$

$\Rightarrow 2^{3x} = 2^2$

$\Rightarrow 3x = 2$

$\Rightarrow x = \dfrac{2}{3}$

Correct Answer: C