First slide

Theory of equations

Question

If $\log_{2} x + \log_{x} 2 = \frac{10}{3} = \log_{2} y + \log_{y} 2$ and $x \neq y$ , then $x + y =$

Moderate

A

2
B

65/8
C

37/6
D

None of these

Solution

We have $\log_{2} x + \frac{1}{\log_{2} x} = 3 + \frac{1}{3} = \log_{2} y + \frac{1}{\log_{2} y}$
$∴ \log_{2} x = 3, \log_{2} y = \frac{1}{3} (∵ x \neq y) \Rightarrow x = 2^{3} and y = 2^{1 / 3} \Rightarrow x + y = 8 + 2^{1 / 3} .$

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