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UPSC CDS I 2025 Mathematics Paper

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Question : 19 of 100

Marks: +1, -0

What is

\log_{10}2000 + \log_{10}400 + 4\log_{10}25 + 5\log_{10}20

equal to?

10
16
18
20

Solution:

Concept:

Use logarithm properties:

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

and

n \log a = \log(a^n)

to combine terms into a single logarithm, then simplify the numeric value.

Explanation:

Start with

\log_{10}2000 + \log_{10}400 + 4\log_{10}25 + 5\log_{10}20

.

Combine all logs:

\log_{10}(2000 \times 400 \times 25^4 \times 20^5)

.

Write each number as powers of 2 and 5:

2000 = 2^4 \times 5^3

,

400 = 2^4 \times 5^2

,

25^4 = (5^2)^4 = 5^8

,

20^5 = (2^2 \times 5)^5 = 2^{10} \times 5^5

.

Multiply:

2^{4+4+10} \times 5^{3+2+8+5} = 2^{18} \times 5^{18} = (2 \times 5)^{18} = 10^{18}

.

So the expression equals

\log_{10}(10^{18}) = 18

.

Answer:

18 (Option C)

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