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

 log _(9/4) ((1/(2(√3)))(√(6−(1/(2(√3)))(√(6−(1/(2(√3)))(√(6−(1/(2(√3)))(√(...))))))))) =?

$$\:\mathrm{log}\:_{\frac{\mathrm{9}}{\mathrm{4}}} \left(\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\sqrt{\mathrm{6}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\sqrt{\mathrm{6}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\sqrt{\mathrm{6}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\sqrt{...}}}}\right)\:=? \\ $$

Answered by EDWIN88 last updated on 18/May/21

let E = (1/(2(√3))) (√(6−(1/(2(√3))) (√(6−(1/(2(√3))) (√(6−(1/(2(√3)))(√…)))))))  ⇒ E = (1/(2(√3))) (√(6−E))   ⇒ E^2  = ((6−E)/(12)) ⇒ 12E^2 +E−6 = 0  ⇒E = ((−1+17)/(24)) = ((16)/(24)) = (2/3)  ∴ log _(((9/4))) ((2/3)) = −(1/2)

$$\mathrm{let}\:\mathcal{E}\:=\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\:\sqrt{\mathrm{6}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\:\sqrt{\mathrm{6}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\:\sqrt{\mathrm{6}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\sqrt{\ldots}}}} \\ $$$$\Rightarrow\:\mathcal{E}\:=\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\:\sqrt{\mathrm{6}−\mathcal{E}}\: \\ $$$$\Rightarrow\:\mathcal{E}^{\mathrm{2}} \:=\:\frac{\mathrm{6}−\mathcal{E}}{\mathrm{12}}\:\Rightarrow\:\mathrm{12}\mathcal{E}^{\mathrm{2}} +\mathcal{E}−\mathrm{6}\:=\:\mathrm{0} \\ $$$$\Rightarrow\mathcal{E}\:=\:\frac{−\mathrm{1}+\mathrm{17}}{\mathrm{24}}\:=\:\frac{\mathrm{16}}{\mathrm{24}}\:=\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\therefore\:\mathrm{log}\:_{\left(\frac{\mathrm{9}}{\mathrm{4}}\right)} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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