Given:
$$\ln \left(\frac{x+y}{2}\right) = \frac{1}{2}(\ln(x) + \ln(y))$$
Step 1: Rewrite the equation.
$$\ln \left(\frac{x+y}{2}\right) = \frac{1}{2}\ln(xy)$$
Step 2: Use the property of logarithms.
$$\ln\left(\frac{x+y}{2}\right) = \ln\left(\sqrt{xy}\right)$$
Step 3: Drop the logarithms.
$$\frac{x+y}{2} = \sqrt{xy}$$
Step 4: Square both sides to eliminate the square root.
$$(x+y)^2 = 4xy$$
$$x^2 + 2xy + y^2 = 4xy$$
$$x^2 + y^2 = 2xy$$
Step 5: Divide both sides by $xy$.
$$\frac{x^2}{xy} + \frac{y^2}{xy} = 2$$
$$\frac{x}{y} + \frac{y}{x} = 2$$
So, the solution to the given equation is $\frac{x}{y} + \frac{y}{x} = 2$. Thus, the correct answer is $2$.
Alternate approach:
Given:
$$\ln \left(\frac{x+y}{2}\right) = \frac{1}{2}(\ln(x) + \ln(y))$$
We observe that when $x = 1$ and $y = 1$, the equation becomes:
$$\ln \left(\frac{1+1}{2}\right) = \frac{1}{2}(\ln(1) + \ln(1))$$
$$\ln(1) = \frac{1}{2}(0 + 0)$$
$$0 = 0$$
So, $x = 1$ and $y = 1$ satisfies the given equation. Substituting these values into $\frac{x}{y} + \frac{y}{x}$, we get:
$$\frac{1}{1} + \frac{1}{1} = 1 + 1 = 2$$
Therefore, according to this alternate method, the answer is $2$.
