Question

The value of \( x (\log y - \log z) \times y (\log z - \log x) \) is equal to:

• A. 0
• B. 3
• C. 5
• D. 1

Hint:

Use logarithmic properties such as \( \log a - \log b = \log \left( \frac{a}{b} \right) \) to simplify expressions involving logarithms.

Answer 1

The Correct Option is A

We are given the expression \( x (\log y - \log z) \times y (\log z - \log x) \). Using logarithmic properties: \[ \log a - \log b = \log \left( \frac{a}{b} \right) \] Thus, the expression becomes: \[ x \log \left( \frac{y}{z} \right) \times y \log \left( \frac{z}{x} \right) \] Multiplying: \[ x \cdot y \cdot \log \left( \frac{y}{z} \right) \cdot \log \left( \frac{z}{x} \right) \] After simplification, it is evident that the expression evaluates to 0. Hence, the correct answer is 0.