2log-x-3-log-3x-3-log-9-x-3-x-

Question Number 161342 by bobhans last updated on 16/Dec/21

2 \log_{x} (3) \log_{3 x} (3) = \log_{9 \sqrt{x}} (3)

x = ?

Answered by 1549442205PVT last updated on 16/Dec/21

t h e g i v e n e q u a t i o n i s e q u i v a l e n t t o

\frac{2}{{l o g}_{3} x} . \frac{1}{1 + {l o g}_{3} x} = \frac{1}{2 + \frac{1}{2} {l o g}_{3} x} (f o r x > 0, x \neq 1)

\Leftrightarrow 2 (4 + {l o g}_{3} x) = 2 . {l o g}_{3} x (1 + {l o g}_{3} x)

\Leftrightarrow {l o g}_{3}^{2} x = 4 \Leftrightarrow {l o g}_{3} x = \pm 2 \Leftrightarrow x = 3^{\pm 2}

\Leftrightarrow x \in {\frac{1}{9}, 9}

Answered by cortano last updated on 16/Dec/21

(Q) 2 \log_{x} (3) \log_{3 x} (3) = \log_{9 \sqrt{x}} (3)

x = ?

(S o l) \frac{2}{\log_{3} (x) . (\log_{3} (x) + 1)} = \frac{1}{\frac{1}{2} \log_{3} (x) + 2}

\frac{2}{\log_{3} (x) (\log_{3} (x) + 1)} = \frac{2}{\log_{3} (x) + 4}

s e t \log_{3} (x) = y

\Leftrightarrow y^{2} + y = y + 4

\Leftrightarrow y = \pm 2 \to {\begin{cases} \log_{3} (x) = 2 \Rightarrow x = 9 \\ \log_{3} (x) = - 2 \Rightarrow x = \frac{1}{9} \end{cases}

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