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: {Substitution} Let \( \log_2 |x| = t \). The equation transforms to \( \sqrt{5 - t} = 3 - t \).
Step 2: {Squaring} Square both sides: \( 5 - t = (3 - t)^2 \). Expanding yields \( 5 - t = 9 + t^2 - 6t \), which simplifies to \( t^2 - 5t + 4 = 0 \).
Step 3: {Solving for t} Factor the quadratic: \( (t - 4)(t - 1) = 0 \). The solutions are \( t = 4 \) or \( t = 1 \). Reject \( t = 4 \) as it is extraneous.
Step 4: {Finding x} Using \( t = 1 \), we have \( \log_2 |x| = 1 \). This means \( |x| = 2 \), so \( x = \pm 2 \).
Step 5: {Conclusion} The real solutions are \( x = 2 \) and \( x = -2 \).