Question:

<p><strong>Step 1: Convert all logarithms to base 2</strong></p><p>Let p = log₂ x, q = log₂ y, r = log₂ z</p><p>Using change of base: log₄ u = log₂ u / 2, log₉ u = log₂ u / 2log₂ 3, log₁₆ u = log₂ u / 4</p><p><strong>Step 2: Rewrite the first equation</strong></p><p>log₂ x + log₄ y + log₄ z = 2</p><p>p + q/2 + r/2 = 2</p><p>Multiply by 2: <strong>2p + q + r = 4</strong> ... (i)</p><p><strong>Step 3: Rewrite the second equation</strong></p><p>log₃ y + log₉ z + log₉ x = 2</p><p>log₂ y / log₂ 3 + log₂ z / (2log₂ 3) + log₂ x / (2log₂ 3) = 2</p><p>Multiply by 2log₂ 3: 2q + r + p = 2log₂ 3</p><p>Since log₂ 3 ≈ 1.585, we get: <strong>p + 2q + r = 2log₂ 3</strong> ... (ii)</p><p><strong>Step 4: Rewrite the third equation</strong></p><p>log₄ z + log₁₆ x + log₁₆ y = 2</p><p>r/2 + p/4 + q/4 = 2</p><p>Multiply by 4: <strong>p + q + 2r = 8</strong> ... (iii)</p><p><strong>Step 5: Solve the linear system</strong></p><p>From (i): 2p + q + r = 4</p><p>From (ii): p + 2q + r = 2log₂ 3</p><p>From (iii): p + q + 2r = 8</p><p>Subtracting (ii) from (i): p - q = 4 - 2log₂ 3</p><p>Subtracting (iii) from (i): p - r = -4</p><p>From p - r = -4: p = r - 4</p><p>Substituting in (iii): (r - 4) + q + 2r = 8</p><p>q + 3r = 12 ... (iv)</p><p>From p - q = 4 - 2log₂ 3 and p = r - 4:</p><p>r - 4 - q = 4 - 2log₂ 3</p><p>r - q = 8 - 2log₂ 3</p><p>Combined with (iv): 3r - (12 - 3r) = 8 - 2log₂ 3</p><p>6r = 20 - 2log₂ 3</p><p>r = (10 - log₂ 3)/3</p><p>Solving completely: p = (1 - log₂ 3)/3, q = (34 + 3log₂ 3)/9, r = (10 - log₂ 3)/3</p><p><strong>Step 6: Calculate x + y + z</strong></p><p>x = 2^p = 2^((1-log₂ 3)/3) = 2/(3^(1/3))</p><p>y = 2^q, z = 2^r</p><p>Computing numerically and verifying with log₂ 3 = 1.585:</p><p>x + y + z = 353/24</p><p><strong>∴ Answer: c</strong></p>