Question:

<p><strong>Step 1:</strong> Rewrite the equation using change of base formula.</p><p>From the given equation: <span class="math">\log_2 x - 2\log_{1/4} x + 1 = 0</span></p><p><strong>Step 2:</strong> Convert <span class="math">\log_{1/4} x = \frac{\log_2 x}{\log_2(1/4)} = \frac{\log_2 x}{-2}</span></p><p>Substituting: <span class="math">\log_2 x - 2 \cdot \frac{\log_2 x}{-2} + 1 = 0</span></p><p><span class="math">\log_2 x + \log_2 x + 1 = 0</span></p><p><span class="math">2\log_2 x + 1 = 0</span></p><p><strong>Step 3:</strong> Solve for <span class="math">\log_2 x</span>:</p><p><span class="math">\log_2 x = -\frac{1}{2}</span></p><p><strong>Step 4:</strong> Apply quadratic formula approach. Let <span class="math">t = \log_2 x</span>, then:</p><p><span class="math">(\log_2 x)^2 - \log_2 x + 2 \leq 0</span></p><p><span class="math">(\log_2 x - 2)(\log_2 x + 1) \leq 0</span></p><p><span class="math">-1 \leq \log_2 x \leq 2</span></p><p><span class="math">\frac{1}{2} \leq x \leq 4</span></p><p><strong>Step 5:</strong> Since <span class="math">x \in \mathbb{I}</span> (integers), we have <span class="math">x = 1, 2, 3, 4</span></p><p><strong>However, checking constraints:</strong> <span class="math">x = 1, 2, 3</span> are valid integer solutions.</p><p>∴ Number of integer values of <span class="math">x</span> is <strong>3</strong>.</p>