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GATE 2026 CE (CE2) – Question 2

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

If a positive real $x$ satisfies the following equation

$$\log_2 x + \log_{\sqrt{2}} x = 48,$$

then the value of $x$ is _________________

  1. $2^{16}$
  2. $4^{16}$
  3. $2^{14}$
  4. $4^{14}$

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Correct answer: (A) $2^{16}$

Explanation

Change the second logarithm to base 2: $\log_{\sqrt{2}} x = \frac{\log_2 x}{\log_2 \sqrt{2}} = \frac{\log_2 x}{1/2} = 2\log_2 x$. The equation becomes $3\log_2 x = 48$, so $\log_2 x = 16$ and $x = 2^{16}$.