First slide

Trigonometric Identities

Question

If the equation $x^{\log_{a} x^{2}} = \frac{x^{k - 2}}{a^{k}}, a \neq 0$ has exactly one solution for r, then the value of k is/are

Moderate

A

$6 + 4 \sqrt{2}$
B

$2 + 6 \sqrt{3}$
C

$6 - 4 \sqrt{2}$
D

$2 - 6 \sqrt{3}$

Solution

$(\log_{a} x^{2}) \log_{a} x = (k - 2) \log_{a} x - k$ (taking log on base a)
let $\log_{a} x = t$ we get
Putting D = 0 (has only one solution), we have
$\begin{array}{ll} (k - 2)^{2} - 8 k = 0 \\ or k^{2} - 12 k + 4 = 0 \\ or k = \frac{12 \pm \sqrt{128}}{2} \\ or k = 6 \pm 4 \sqrt{2} \end{array}$

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