What is ₁0}31.25 equal to?

Correct Answer is : 3−5log⁡1023 - 5 ₁0}23−5log102

Correct Answer is : 3−5log⁡1023 - 5 ₁0}23−5log102

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UPSC CDS 1 2021 Math Paper

Question : 51 of 100

Marks: +1, -0

What is

\log_{10}31.25

equal to?

Solution:

Concept:

Use properties of logarithms to express

31.25

as a fraction of powers of

2

and

5

, then simplify using

\log_{10}5 = 1 - \log_{10}2

Explanation:

First, write

31.25

as a fraction:

31.25 = \frac{3125}{100}

Now,

3125 = 5^5

and

100 = 10^2

. So,

\log_{10}31.25 = \log_{10}\frac{5^5}{10^2}

Using

\log_{10}\frac{a}{b} = \log_{10}a - \log_{10}b

, we get:

\log_{10}5^5 - \log_{10}10^2

Apply

\log_{10}a^n = n\log_{10}a

5\log_{10}5 - 2\log_{10}10

Since

\log_{10}10 = 1

, this becomes

5\log_{10}5 - 2

Now,

\log_{10}5 = \log_{10}\frac{10}{2} = \log_{10}10 - \log_{10}2 = 1 - \log_{10}2

Substitute:

5(1 - \log_{10}2) - 2 = 5 - 5\log_{10}2 - 2 = 3 - 5\log_{10}2

Thus,

\log_{10}31.25

simplifies to

3 - 5\log_{10}2

Answer:

Option A:

3 - 5\log_{10}2

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