Question

For positive non-zero real variables \( x \) and \( y \), if \[ \ln \left( \frac{x + y}{2} \right) = \frac{1}{2} \left[ \ln(x) + \ln(y) \right], \] then, the value of \( \frac{x}{y} + \frac{y}{x} \) is:

• A. ( 1 )
• B. ( frac{1}{2} )
• C. ( 2 )
• D. ( 4 )

Hint:

When solving logarithmic equations, simplify step-by-step using logarithmic properties, and always verify the solution by back-substitution.

Answer 1

The Correct Option is C

Step 1: Simplify the given logarithmic equation.

The given equation is:

\[ \ln \left( \frac{x + y}{2} \right) = \frac{1}{2} \left[ \ln(x) + \ln(y) \right]. \]

Using the logarithm property, \( \ln(ab) = \ln(a) + \ln(b) \), we write:

\[ \frac{1}{2} \left[ \ln(x) + \ln(y) \right] = \frac{1}{2} \ln(xy). \]

Thus, the equation becomes:

\[ \ln \left( \frac{x + y}{2} \right) = \frac{1}{2} \ln(xy). \] Step 2: Exponentiate both sides.

Exponentiating both sides, we get:

\[ \frac{x + y}{2} = \sqrt{xy}. \]

Multiplying through by 2:

\[ x + y = 2\sqrt{xy}. \] Step 3: Divide through by \( \sqrt{xy} \).

Dividing both sides by \( \sqrt{xy} \), we get:

\[ \frac{x}{\sqrt{xy}} + \frac{y}{\sqrt{xy}} = 2. \]

Simplify the terms:

\[ \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 2. \] Step 4: Square both sides.

Squaring both sides:

\[ \left( \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} \right)^2 = 2^2. \]

Expanding the square:

\[ \frac{x}{y} + \frac{y}{x} + 2 = 4. \] Step 5: Solve for \( \frac{x}{y} + \frac{y}{x} \).

Subtracting 2 from both sides:

\[ \frac{x}{y} + \frac{y}{x} = 2. \]

Thus, the value of \( \frac{x}{y} + \frac{y}{x} \) is \( \boxed{2} \).