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UPSC CDS 1 2020 Math Paper

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Question : 3 of 100

Marks: +1, -0

How many digits are there in

(54)^{10}

\log_{10}2 = 0.301

\log_{10}3 = 0.477

)

16
18
19
27

Solution:

Concept:

The number of digits in any positive number

n

is given by

\lfloor \log_{10} n \rfloor + 1

, where

\lfloor x \rfloor

is the floor function (largest integer ≤

x

).

Explanation:

We need to find the number of digits in

(54)^{10}

.

First, compute

\log_{10} (54)^{10}

.

Using the power rule:

\log_{10} (54)^{10} = 10 \times \log_{10} 54

.

Now factor

54 = 2 \times 3^3

.

So

\log_{10} 54 = \log_{10} (2 \times 3^3) = \log_{10} 2 + 3 \log_{10} 3

.

Substitute given values:

\log_{10} 2 = 0.301

and

\log_{10} 3 = 0.477

.

Thus,

3 \log_{10} 3 = 3 \times 0.477 = 1.431

.

Therefore,

\log_{10} 54 = 0.301 + 1.431 = 1.732

.

Then

\log_{10} (54)^{10} = 10 \times 1.732 = 17.32

.

Take floor:

\lfloor 17.32 \rfloor = 17

.

Number of digits =

17 + 1 = 18

.

Hence,

(54)^{10}

has 18 digits.

Answer:

B. 18

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