Question

The solution of the equation \( \log\left(\log_4(\sqrt{x+4} + \sqrt{x})\right) = 0 \) is:

• A. \(2\)
• B. \(4\)
• C. \( \frac{9}{4} \)
• D. \(8\)

Hint:

Always verify solution after squaring to avoid extraneous roots.

Answer 1

The Correct Option is C

Concept: \[ \log y = 0 \Rightarrow y = 1 \]

Step 1: \[ \log\left(\log_4(\sqrt{x+4} + \sqrt{x})\right) = 0 \Rightarrow \log_4(\sqrt{x+4} + \sqrt{x}) = 1 \]

Step 2: \[ \sqrt{x+4} + \sqrt{x} = 4^1 = 4 \]

Step 3: Let \( \sqrt{x} = t \Rightarrow \sqrt{x+4} = \sqrt{t^2 + 4} \) \[ \sqrt{t^2 + 4} + t = 4 \]

Step 4: \[ \sqrt{t^2 + 4} = 4 - t \] Squaring: \[ t^2 + 4 = (4 - t)^2 = 16 - 8t + t^2 \] \[ 4 = 16 - 8t \Rightarrow 8t = 12 \Rightarrow t = \frac{3}{2} \]

Step 5: \[ x = t^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \] But check: \[ \sqrt{x+4} + \sqrt{x} = \sqrt{\frac{9}{4} + 4} + \frac{3}{2} = \sqrt{\frac{25}{4}} + \frac{3}{2} = \frac{5}{2} + \frac{3}{2} = 4 \] \[ \log_4(4) = 1 \Rightarrow \log(A) = 0 \] Final: \[ {\frac{9}{4}} \]