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UPSC NDA II 2015 Mathematics Paper

Question : 40 of 120

Marks: +1, -0

f(xy) = f(x) + f(y) is true for all

Solution:

Concept:

The functional equation

f(xy) = f(x) + f(y)

is the defining property of logarithmic functions.

Explanation:

Let

f(x) = \log_b x

for some base

b > 0, b \neq 1

Then

f(xy) = \log_b (xy)

By the product rule of logarithms:

\log_b (xy) = \log_b x + \log_b y

Thus

f(xy) = \log_b x + \log_b y = f(x) + f(y)

No polynomial, trigonometric, or exponential function satisfies this equation for all inputs.

Answer:

D. Logarithmic function f

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