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UPSC CDS 2 2021 Math Paper

Question : 13 of 100

Marks: +1, -0

\log_{10}x + \log_{10}x^{2} = 2\log_{10}x + 1

x

Solution:

Concept:

Use logarithm properties to combine terms, then solve the resulting algebraic equation.

Explanation:

Given:

\log_{10}x + \log_{10}x^{2} = 2\log_{10}x + 1

Apply

\log a + \log b = \log(ab)

on the left side:

\log_{10}(x \cdot x^{2}) = \log_{10}x^{3}

Write

1

\log_{10}10

and

2\log_{10}x

\log_{10}x^{2}

on the right side:

\log_{10}x^{2} + \log_{10}10 = \log_{10}(10x^{2})

Now the equation becomes

\log_{10}x^{3} = \log_{10}(10x^{2})

Since the bases are equal, equate the arguments:

x^{3} = 10x^{2}

Rearrange:

x^{3} - 10x^{2} = 0 \;\Rightarrow\; x^{2}(x - 10) = 0

Thus

x = 0

x = 10

Logarithm of zero is undefined, so

x = 0

is rejected.

Hence

x = 10

Answer:

x = 10

(Option D)

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