The GATE Grind

GATE 2023 CE (CE2) – Question 9

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

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

  1. $1/2$
  2. $\log_{16}8$
  3. $2/3$
  4. $\log_48$

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Correct answer: (C) $2/3$

Explanation

**Step 1: write 256 as a power of 4.**
$$256=4^4 .$$

**Step 2: equate exponents** (same base):
$$4^{8^x}=4^4\;\Rightarrow\;8^x=4 .$$

**Step 3: write both sides with base 2.**
$$8^x=(2^3)^x=2^{3x},\qquad 4=2^2\;\Rightarrow\;3x=2\;\Rightarrow\;x=\frac23 .$$

Check: $8^{2/3}=(8^{1/3})^2=2^2=4$ ✓. Answer **C**.