For positive non-zero real variables \(x\) and \(y\), if
\[\ln\left(\frac{x + y}{2}\right) = \frac{1}{2}[\ln(x) + \ln(y)]\]
then, the value of \(\frac{x}{y} + \frac{y}{x}\) is
Answer
(C)
Explanation
The right side is a single logarithm: \(\frac{1}{2}[\ln(x) + \ln(y)] = \ln\sqrt{xy}\). Since \(\ln\) is one-to-one,
\[\frac{x + y}{2} = \sqrt{xy} \implies x - 2\sqrt{xy} + y = 0 \implies (\sqrt{x} - \sqrt{y})^{2} = 0.\]
So \(x = y\), the only case in which the arithmetic mean equals the geometric mean, and \(\frac{x}{y} + \frac{y}{x} = 1 + 1 = 2\).
Options 1 and 1/2 could never occur: for positive \(x\) and \(y\), \(\frac{x}{y} + \frac{y}{x}\) is never less than 2.
Answer: C