First slide

Introduction to sets

Question

The value of $\log_{\frac{9}{4}} (\frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}}}}} \dots \infty) is$

Easy

A

-2
B

-1
C

$- \frac{1}{2}$
D

None of these

Solution

$\begin{array}{l} Let x = \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}}}}} \dots \infty \\ \Rightarrow x = \frac{1}{2 \sqrt{3}} \sqrt{6 - x} \\ \Rightarrow 12 x^{2} + x - 6 = 0 \\ \Rightarrow (3 x - 2) (4 x + 3) = 0 \\ \Rightarrow x = \frac{2}{3} \end{array}$
$\begin{array}{l} \Rightarrow \log_{\frac{9}{4}} (\frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}} \sqrt{6 - \frac{1}{2 \sqrt{3}}}}} \dots \infty) \\ = \log_{\frac{9}{4}} (\frac{2}{3}) = - \frac{1}{2} \end{array}$

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