log-9-4-1-2-3-6-1-2-3-6-1-2-3-6-1-2-3-

Question Number 141365 by bemath last updated on 18/May/21

\log_{\frac{9}{4}} (\frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{\dots}}}}) = ?

Answered by EDWIN88 last updated on 18/May/21

let E = \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{\dots}}}}

\Rightarrow E = \frac{1}{2 \sqrt{3}} \sqrt{6 - E}

\Rightarrow E^{2} = \frac{6 - E}{12} \Rightarrow 12 E^{2} + E - 6 = 0

\Rightarrow E = \frac{- 1 + 17}{24} = \frac{16}{24} = \frac{2}{3}

∴ \log_{(\frac{9}{4})} (\frac{2}{3}) = - \frac{1}{2}

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