The correct answer is **B**. Let the given expression be $E$. $$ E = \frac{1}{\log_{xy} (xyz)} + \frac{1}{\log_{yz} (xyz)} + \frac{1}{\log_{zx} (xyz)} $$ **Step 1: Apply the change of base property for reciprocals.** We know that for any positive real numbers $a, b$ where $a, b \neq 1$, the property $\frac{1}{\log_a b} = \log_b a$ holds true. Applying this property to each term in the expression $E$: * For the first term: $\frac{1}{\log_{xy} (xyz)} = \log_{xyz} (xy)$ * For the second term: $\frac{1}{\log_{yz} (xyz)} = \log_{xyz} (yz)$ * For the third term: $\frac{1}{\log_{zx} (xyz)} = \log_{xyz} (zx)$ Substituting these into the expression $E$, we get: $$ E = \log_{xyz} (xy) + \log_{xyz} (yz) + \log_{xyz} (zx) $$ **Step 2: Apply the logarithm product rule.** We know that for any positive real numbers $a, M, N$ where $a \neq 1$, the property $\log_a M + \log_a N = \log_a (M \cdot N)$ holds true. Applying this property to combine the three terms, since they all have the same base ($xyz$): $$ E = \log_{xyz} (xy \cdot yz \cdot zx) $$ **Step 3: Simplify the argument of the logarithm.** Multiply the terms inside the logarithm: $$ xy \cdot yz \cdot zx = (x \cdot x) \cdot (y \cdot y) \cdot (z \cdot z) = x^2 y^2 z^2 $$ So, the expression becomes: $$ E = \log_{xyz} (x^2 y^2 z^2) $$ **Step 4: Rewrite the argument as a power of the base.** The term $x^2 y^2 z^2$ can be written as $(xyz)^2$. Thus, the expression is: $$ E = \log_{xyz} ((xyz)^2) $$ **Step 5: Apply the logarithm power rule.** We know that for any positive real numbers $a, M$ where $a \neq 1$, and any real number $k$, the property $\log_a (M^k) = k \log_a M$ holds true. Applying this property: $$ E = 2 \log_{xyz} (xyz) $$ **Step 6: Evaluate the final logarithm.** We know that for any positive real number $a$ where $a \neq 1$, the property $\log_a a = 1$ holds true. Since the base and the argument are both $xyz$ (and $xyz \neq 1$ as per the problem statement): $$ \log_{xyz} (xyz) = 1 $$ Therefore, $$ E = 2 \cdot 1 = 2 $$ The final value of the expression is $2$.