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UPSC NDA II 2018 Mathematics Solved Paper

Question : 17 of 120

Marks: +1, -0

\log_{9}27 + \log_{8}32

Solution:

Concept:

Use the change of base formula and power rule of logarithms to simplify each term.

Explanation:

First, rewrite each logarithm using base 10:

\log_{9}27 = \frac{\log 27}{\log 9}

and

\log_{8}32 = \frac{\log 32}{\log 8}

Now express numbers as powers of primes:

27 = 3^{3}

9 = 3^{2}

32 = 2^{5}

8 = 2^{3}

Apply the power rule:

\log a^{b} = b \log a

Thus,

\frac{\log 3^{3}}{\log 3^{2}} = \frac{3\log 3}{2\log 3} = \frac{3}{2}

Similarly,

\frac{\log 2^{5}}{\log 2^{3}} = \frac{5\log 2}{3\log 2} = \frac{5}{3}

Add the fractions:

\frac{3}{2} + \frac{5}{3} = \frac{9+10}{6} = \frac{19}{6}

Answer:

\frac{19}{6}

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