First slide

Theory of equations

Question

Sum of the roots of the equation $9^{\log_{3}}^{(\log_{2} x)} = \log_{2} x - {(\log_{2} x)}^{2} + 1$ is:

Moderate

A

$1 / 2$
B

$2 + 1 / \sqrt{2}$
C

$\sqrt{2}$
D

$2$

Solution

$9^{\log_{3}}^{(\log_{2} x)} = \log_{2} x - {(\log_{2} x)}^{2} + 1$
$\Rightarrow$    $3^{\log_{3} {(\log_{2} x)}^{2}} = \log_{2} x - {(\log_{2} x)}^{2} + 1$

$\Rightarrow$    ${(\log_{2} x)}^{2} = \log_{2} x - {(\log_{2} x)}^{2} + 1$                           $(\log_{2} x > 0)$

$\Rightarrow$    $2 {(\log_{2} x)}^{2} - \log_{2} x - 1 = 0$

$\Rightarrow$    $\log_{2} x = - 1 / 2$     or $\log_{2} x = 1$

As $\log_{2} x > 0, \log_{2} x = 1$    or $x = 2$

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