First slide

Trigonometric Identities

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

$\frac{10}{3} = 3 + \frac{1}{3} . The given equation is of the form$
$p + \frac{1}{p} = 3 + \frac{1}{3} = q + \frac{1}{q}, where p \neq q as x \neq y .$
$\begin{array}{ll} \Rightarrow \log_{2} x = 3, \log_{2} y = 1 / 3 \\ \Rightarrow x = 2^{3}, y = 2^{1 / 3} \\ ∴ x + y = 8 + 2^{1 / 3} \end{array}$

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