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

Question

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

Accepted Answer

Correct answer: C — Key idea: exponentiate the logarithmic equality to relate the arithmetic and geometric means. Start with the given equality: ln((x + y)/2) = (1/2)[ln x + ln y]. Exponentiate both sides to remove the logarithms: (x + y)/2 = sqrt(xy). Rearrange: x + y = 2 sqrt(xy) which gives x - 2 sqrt(xy) + y = 0, i.e. (sqrt x - sqrt y)^2 = 0. Thus sqrt x = sqrt y, so x = y (since x and y are positive). Finally compute the required expression: x/y + y/x = 1 + 1 = 2. Answer: 2

Answer

A. 1

Answer

B. 1/2

Answer

C. 2

Answer

D. 4

For positive non-zero real variables 𝑥 and 𝑦, if \(\ln\left(\frac{x +…