Question:
The number of real solutions of
\[
\sqrt{5 - \log_2 |x|} = 3 - \log_2 |x|
\]
is:
The number of real solutions of
\[ \sqrt{5 - \log_2 |x|} = 3 - \log_2 |x| \] is:
\[ \sqrt{5 - \log_2 |x|} = 3 - \log_2 |x| \] is:
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When solving logarithmic equations, check that all solutions satisfy the original equation.
Updated On: Feb 4, 2025
- \( 1 \)
- \( 2 \)
- \( 3 \)
- \( 4 \)
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The Correct Option is B
Solution and Explanation
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: \( x = 2, -2 \).
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