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UPSC NDA Math Model Paper 5

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Question : 107 of 120

Marks: +1, -0

What is the value of

\log_{3} \log_{3} \sqrt{3 \sqrt{3}}

equal to?

$3 \log_{2}(3)$
$1 - 3 \log_{2}(2)$
$1 - 2 \log_{3}(2)$
$\frac{3}{8}$

Solution:

Here, we have to find the value of

\log_{3} \log_{3} \sqrt{3 \sqrt{3}}

Now,

\log_{3} \log_{3} \sqrt{3 \sqrt{3}} = \log_{3} \log_{3}\left(3^{\frac{1}{2}} \times 3^{\frac{1}{4}}\right)

= \log_{3} \log_{3}\left(3^{\left(\frac{1}{2}+\frac{1}{4}\right)}\right)

= \log_{3} \log_{3}\left(3^{\frac{2+1}{4}}\right)

= \log_{3} \log_{3}\left(3^{\frac{3}{4}}\right)

From power rule;

= \log_{3} \left[ \frac{3}{4} \times \log_{3} 3 \right] \left[ \because \log_{a}(m)^{n} = n \times \log_{a}(m) \right]

= \log_{3} \left( \frac{3}{4} \right) (\because \log_{m} m = 1)

= \log_{3} \left( \frac{3}{4} \right) = \log_{3} 3 - \log_{3} 4

= 1 - \log_{3} 4 = 1 - \log_{3} 2^{2}

= 1 - 2 \log_{3} 2

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