Question-14

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

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NoteAnswer

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NoteSoluton

\[ \begin{array}{ c r l } & \ln\left(\frac{x+y}{2}\right) & =\frac{1}{2}(\ln x+\ln y)\\ \Longrightarrow & \ln\left(\frac{x+y}{2}\right) & =\frac{1}{2}\ln xy\\ \Longrightarrow & 2\ln\left(\frac{x+y}{2}\right) & =\ln xy\\ \Longrightarrow & \ln\left[\left(\frac{x+y}{2}\right)^{2}\right] & =\ln xy\\ \Longrightarrow & \ln\left[\frac{1}{xy} \cdot \left(\frac{x+y}{2}\right)^{2}\right] & =0\\ \Longrightarrow & \frac{1}{xy} \cdot \left(\frac{x+y}{2}\right)^{2} & =1\\ \Longrightarrow & \frac{x}{y} +\frac{y}{x} & =2 \end{array} \]