Question

If $x \geq y>1$, the value of $\log_x\left(\frac{x}{y}\right) + \log_y\left(\frac{y}{x}\right)$ can never be:

• A. -1
• B. -0.5
• C. 0
• D. 1

Hint:

Transform logs to a single base, then apply inequalities for sum bounds.

Answer 1

The Correct Option is A

Let $t = \log_x y \in (0,1]$. Expression becomes $\frac{1}{t} - 1 + t - 1$. By AM ≥ GM, $t + \frac{1}{t} \geq 2$, so min value = 0. Thus -1 impossible. \[ \boxed{-1} \]