<p>If \(a, b, c, d \in R^+ - \{1\}\), then the minimum value of \(\log_a d + \log_b d + \log_c d + \log_a c + \log_b c\) is</p>
Step-by-Step Solution
Key Concept: Convert all logarithms to a common base using the change of base formula, then express in terms of reciprocals: log_a(d) = 1/log_d(a). Use AM-GM inequality on the resulting expression with constraint optimization.
<p><strong>Step 1:</strong> Use change of base formula to convert to natural logarithm:</p><p>log_a(d) = ln(d)/ln(a), log_b(d) = ln(d)/ln(b), log_c(d) = ln(d)/ln(c), log_a(c) = ln(c)/ln(a), log_b(c) = ln(c)/ln(b)</p><p><strong>Step 2:</strong> Let x = ln(a), y = ln(b), z = ln(c), w = ln(d) where x,y,z,w > 0 and ≠ 0</p><p>Expression becomes: w/x + w/y + w/z + z/x + z/y</p><p><strong>Step 3:</strong> Factor as: w(1/x + 1/y + 1/z) + z(1/x + 1/y)</p><p><strong>Step 4:</strong> Apply AM-GM inequality: (1/x + 1/y + 1/z) ≥ 3∛(1/xyz) and (1/x + 1/y) ≥ 2√(1/xy)</p><p><strong>Step 5:</strong> For minimum, set w = z (equivalently d = c) and x = y = z (equivalently a = b = c)</p><p><strong>Step 6:</strong> Substituting optimum values: w(3/x) + w(2/x) = 5w/x = 5</p><p>∴ Answer: A (minimum value is <strong>5</strong>)</p>
Correct Answer: A