Question

The number of real solutions of
\[ \sqrt{5 - \log_2 |x|} = 3 - \log_2 |x| \] is:

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

When solving logarithmic equations, check that all solutions satisfy the original equation.

Answer 1

The Correct Option is B

Step 1: {Substituting \( \log_2 |x| = t \)} 
Let \[ \log_2 |x| = t. \] Thus, the equation becomes: \[ \sqrt{5 - t} = 3 - t. \] 
Step 2: {Squaring both sides} 
\[ 5 - t = (3 - t)^2. \] Expanding: \[ 5 - t = 9 + t^2 - 6t. \] \[ t^2 - 5t + 4 = 0. \] 
Step 3: {Solving for \( t \)} 
\[ (t - 4)(t - 1) = 0. \] \[ t = 4 \quad {or} \quad t = 1. \] Rejecting \( t = 4 \) as it violates the equation. 
Step 4: {Finding \( x \)} 
\[ \log_2 |x| = 1. \] \[ |x| = 2. \] \[ x = \pm 2. \] 
Step 5: {Conclusion} 
Thus, there are \( 2 \) real solutions: \( x = 2, -2 \).

Answer 2

Step 1: Simplify the equation
We are given the equation: \[ \sqrt{5 - \log_2 |x|} = 3 - \log_2 |x|. \] Let \( y = \log_2 |x| \). Then the equation becomes: \[ \sqrt{5 - y} = 3 - y. \] Step 2: Square both sides
Square both sides to eliminate the square root: \[ 5 - y = (3 - y)^2. \] Expanding the right-hand side: \[ 5 - y = 9 - 6y + y^2. \] Now, rearrange the terms: \[ 0 = y^2 - 5y + 4. \] Step 3: Solve the quadratic equation
We now have a quadratic equation: \[ y^2 - 5y + 4 = 0. \] Solve this using the quadratic formula: \[ y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(4)}}{2(1)} = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm \sqrt{9}}{2} = \frac{5 \pm 3}{2}. \] So, the two solutions for \( y \) are: \[ y = \frac{5 + 3}{2} = 4 \quad \text{or} \quad y = \frac{5 - 3}{2} = 1. \] Step 4: Solve for \( x \)
Recall that \( y = \log_2 |x| \), so we have the equations: \[ \log_2 |x| = 4 \quad \text{or} \quad \log_2 |x| = 1. \] From \( \log_2 |x| = 4 \), we get: \[ |x| = 2^4 = 16 \quad \Rightarrow \quad x = \pm 16. \] From \( \log_2 |x| = 1 \), we get: \[ |x| = 2^1 = 2 \quad \Rightarrow \quad x = \pm 2. \] Step 5: Conclusion
Thus, the four possible solutions for \( x \) are \( x = \pm 16 \) and \( x = \pm 2 \). However, we must check these solutions in the original equation to ensure they satisfy it. 
Verification:
For \( x = 16 \): \[ \sqrt{5 - \log_2 16} = \sqrt{5 - 4} = \sqrt{1} = 1 \quad \text{and} \quad 3 - \log_2 16 = 3 - 4 = -1. \] So, \( x = 16 \) does not satisfy the equation. For \( x = -16 \), the same calculation applies, so \( x = -16 \) does not satisfy the equation. For \( x = 2 \): \[ \sqrt{5 - \log_2 2} = \sqrt{5 - 1} = \sqrt{4} = 2 \quad \text{and} \quad 3 - \log_2 2 = 3 - 1 = 2. \] So, \( x = 2 \) satisfies the equation. For \( x = -2 \), the same calculation applies, so \( x = -2 \) satisfies the equation. 
Final Answer:
The number of real solutions is:

2