log-x-6-2-log-1-6-1-x-2-log-1-x-1-6-log-6-x-3-4-0-

Question Number 89456 by jagoll last updated on 17/Apr/20

{(\log_{x} (6))}^{2} + {(\log_{\frac{1}{6}} (\frac{1}{x}))}^{2} +

\log_{\frac{1}{\sqrt{x}}} (\frac{1}{6}) + \log_{\sqrt{6}} (x) + \frac{3}{4} = 0

Answered by john santu last updated on 17/Apr/20

(1) {(\log_{\frac{1}{6}} (\frac{1}{x}))}^{2} = {(\log_{6} (x))}^{2}

(2) \log_{\frac{1}{\sqrt{x}}} (\frac{1}{6}) = 2 \log_{x} (6)

(3) \log_{\sqrt{6}} (x) = 2 \log_{6} (x)

\Rightarrow l e t \log_{6} (x) = t

t^{2} + \frac{1}{t^{2}} + 2 t + \frac{2}{t} + \frac{3}{4} = 0

{(t + \frac{1}{t})}^{2} + 2 (t + \frac{1}{t}) - \frac{5}{4} = 0

\Rightarrow 4 u^{2} + 8 u - 5 = 0

(2 u - 1) (2 u + 5) = 0

(i) 2 t + \frac{2}{t} - 1 = 0

2 t^{2} - t + 2 = 0, t \notin R

(i i) 2 t + \frac{2}{t} + 5 = 0

2 t^{2} + 5 t + 2 = 0

(2 t + 1) (t + 2) = 0

{\begin{cases} \log_{6} (x) = - 2 \Rightarrow x = \frac{1}{36} \\ \log_{6} (x) = - \frac{1}{2} \Rightarrow x = \frac{1}{\sqrt{6}} \end{cases}

Commented by jagoll last updated on 17/Apr/20

g r e a t s i r . t h a n k y o u

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