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UPSC NDA II 2018 Mathematics Solved Paper

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Question : 1 of 120

Marks: +1, -0

What is the value of

log_{7} log_{7}\sqrt{7\sqrt{7\sqrt{7}}}

equal to?

$3 log_{2} 7$
1 – 3 log2 7
1 – 3 log7 2
$\frac{7}{8}$

Solution:

Concept:

Use logarithm rules: power rule

\log_a(m^n) = n \log_a m

, product rule

\log_a(mn) = \log_a m + \log_a n

, and

\log_a a = 1

.

Explanation:

Simplify the inner expression:

\sqrt{7\sqrt{7\sqrt{7}}} = 7^{1/2} \times 7^{1/4} \times 7^{1/8} = 7^{1/2 + 1/4 + 1/8} = 7^{(4+2+1)/8} = 7^{7/8}

.

Now compute

\log_7\sqrt{7\sqrt{7\sqrt{7}}} = \log_7(7^{7/8}) = \frac{7}{8} \log_7 7 = \frac{7}{8}

.

Then the given expression becomes

\log_7\left(\frac{7}{8}\right)

.

Apply quotient rule:

\log_7 7 - \log_7 8 = 1 - \log_7 8

.

Since

8 = 2^3

, use power rule:

\log_7 8 = 3 \log_7 2

.

Thus final value is

1 - 3 \log_7 2

.

Answer:

1 - 3 \log_7 2

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