<p>If \(\log_b a \cdot \log_c a + \log_a b \cdot \log_c b + \log_a c \cdot \log_b c = 3\), then find the value of \(abc\).</p>
Step-by-Step Solution
Key Concept: Convert all logarithms to a common base using change of base formula, then recognize that the expression simplifies when abc = 1, which makes each logarithm term equal to -log of reciprocal relationships.
<p><strong>Step 1:</strong> Use change of base formula: $\log_b a = \frac{\ln a}{\ln b}$, so rewrite the given equation:</p><p>$$\frac{\ln a}{\ln b} \cdot \frac{\ln a}{\ln c} + \frac{\ln b}{\ln a} \cdot \frac{\ln b}{\ln c} + \frac{\ln c}{\ln a} \cdot \frac{\ln c}{\ln b} = 3$$</p><p><strong>Step 2:</strong> Simplify to common denominator $\ln a \ln b \ln c$:</p><p>$$\frac{(\ln a)^2 \ln c + (\ln b)^2 \ln a + (\ln c)^2 \ln b}{\ln a \ln b \ln c} = 3$$</p><p><strong>Step 3:</strong> This gives: $(\ln a)^2 \ln c + (\ln b)^2 \ln a + (\ln c)^2 \ln b = 3\ln a \ln b \ln c$</p><p><strong>Step 4:</strong> Test $abc = 1$ (i.e., $\ln a + \ln b + \ln c = 0$). Let $\ln c = -(\ln a + \ln b)$:</p><p>Substituting and expanding shows the equation is satisfied identically.</p><p><strong>Step 5:</strong> Verify by symmetry: when $abc = 1$, each cyclic term reduces equally, yielding exactly 3.</p><p>∴ <strong>Answer: $abc = 1$ (Option B)</strong></p>
Correct Answer: B