Question

If \(\log{\left(\frac{x+y}{3}\right)} = \frac{1}{2} \left(\log{x} + \log{y}\right)\) then the value of \(\frac{x}{y} + \frac{y}{x}\) is?

• A. 7
• B. 8
• C. -7
• D. 9

Hint:

To solve logarithmic equations, use properties such as \(\log{a} + \log{b} = \log{(ab)}\) and \(\frac{1}{2} \log{a} = \log{\sqrt{a}}\) to simplify the expressions.

Answer 1

The Correct Option is A

Step 1: Given equation.
The given equation is \[ \log{\left(\frac{x+y}{3}\right)} = \frac{1}{2} \left(\log{x} + \log{y}\right). \]

Step 2: Simplify the equation.
Using the logarithmic identity \(\log{a} + \log{b} = \log{(ab)}\), we can rewrite the equation as: \[ \log{\left(\frac{x+y}{3}\right)} = \frac{1}{2} \log{(xy)}. \] Now, use the property \(\frac{1}{2} \log{(xy)} = \log{\sqrt{xy}}\) to obtain: \[ \log{\left(\frac{x+y}{3}\right)} = \log{\sqrt{xy}}. \]

Step 3: Equate the arguments.
Since the logarithms are equal, their arguments must also be equal. Thus: \[ \frac{x+y}{3} = \sqrt{xy}. \] Multiplying both sides by 3: \[ x + y = 3\sqrt{xy}. \]

Step 4: Find the value of \(\frac{x}{y} + \frac{y}{x}\).
Now, square both sides of the equation \(x + y = 3\sqrt{xy}\) to obtain: \[ (x + y)^2 = 9xy. \] Expanding the left-hand side: \[ x^2 + 2xy + y^2 = 9xy. \] Rearranging the terms: \[ x^2 + y^2 = 7xy. \] Now, divide both sides by \(xy\): \[ \frac{x^2}{xy} + \frac{y^2}{xy} = 7. \] Simplifying: \[ \frac{x}{y} + \frac{y}{x} = 7. \]

Final Answer: 7.