<p>If \(-4 \leq x < 2\), then \(\|x+2|-3|\) lies in the interval</p>
Step-by-Step Solution
Key Concept: To find the range of a logarithmic function, apply logarithm properties to the inequality constraint, recognizing that log is an increasing function so the inequality direction is preserved.
<p><strong>Step 1:</strong> Given constraint: <em>-4 ≤ x < -1</em></p><p><strong>Step 2:</strong> Since we need log(2<sup>x</sup>), note that 2<sup>x</sup> > 0 for all real x, so the logarithm is defined.</p><p><strong>Step 3:</strong> When <em>-4 ≤ x < -1</em>, raise 2 to these powers: <em>2<sup>-4</sup> ≤ 2<sup>x</sup> < 2<sup>-1</sup></em></p><p><strong>Step 4:</strong> This gives: <em>1/16 ≤ 2<sup>x</sup> < 1/2</em></p><p><strong>Step 5:</strong> Apply log₂ (or common/natural log) to all parts. Since log is an increasing function, inequality direction is preserved:</p><p><em>log₂(1/16) ≤ log₂(2<sup>x</sup>) < log₂(1/2)</em></p><p><strong>Step 6:</strong> Simplify: <em>-4 ≤ x < -1</em> (if using log₂)</p><p>Alternatively, using natural log: <em>ln(1/16) ≤ x·ln(2) < ln(1/2)</em></p><p>Which gives: <em>-4ln(2) ≤ x·ln(2) < -ln(2)</em>, so <em>-4 ≤ x < -1</em></p><p><strong>Answer: C</strong> (Range is <em>[-4ln(2), -ln(2))</em> or equivalent interval depending on logarithm base used)</p>
Correct Answer: C