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UPSC NDA I 2019 Mathematics Solved Paper

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Question : 72 of 120

Marks: +1, -0

If

f(x) = \log_{10} (1 + x)

4f(4) + 5f(1) - \log_{10} 2

equal to?

0
1
2
4

Solution:

Concept:

Use the property

a \log_b c = \log_b (c^a)

and combine logarithms to simplify the expression.

Explanation:

Given

f(x) = \log_{10} (1 + x)

.

Compute

f(4) = \log_{10} (5)

and

f(1) = \log_{10} (2)

.

Then

4f(4) = 4\log_{10} (5) = \log_{10} (5^4) = \log_{10} (625)

.

Similarly,

5f(1) = 5\log_{10} (2) = \log_{10} (2^5) = \log_{10} (32)

.

So the given expression becomes

\log_{10} (625) + \log_{10} (32) - \log_{10} (2)

.

Combine the first two using

\log A + \log B = \log(AB)

:

\log_{10} (625 \times 32) = \log_{10} (20000)

.

Now subtract

\log_{10} 2

:

\log_{10} (20000) - \log_{10} (2) = \log_{10} \left( \frac{20000}{2} \right) = \log_{10} (10000)

.

Since

10000 = 10^4

, we have

\log_{10} (10^4) = 4

.

Thus the value is

4

.

Answer:

4

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