If x=10 g₃ 5, y=₁7} 25, which one of the following is correct ?

Correct Answer is : x>yx > yx>y

Correct Answer is : x>yx > yx>y

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Logarithms

Question : 34 of 85

Marks: +1, -0

x=10 g_{3} 5, y=\log_{17} 25,

which one of the following is correct ?

Solution:

Concept:

Compare the values of

x = 10 \log_3 5

and

y = \log_{17} 25

using logarithm properties and approximate magnitudes.

Explanation:

First, simplify

y

y = \log_{17} 25 = \log_{17} (5^2) = 2 \log_{17} 5

Now,

x = 10 \log_3 5

. Since

\log_3 5

is between 1 and 2 (because

3^1=3

3^2=9

), we have

x

between 10 and 20.

The value

\log_{17} 5

is less than 1 (since

17^1=17 > 5

), so

y = 2\log_{17}5

is less than 2.

Thus

x > 10

and

y < 2

, clearly

x > y

Alternatively, using change of base:

x / y = (10 \log_3 5) / (2 \log_{17} 5) = 5 \cdot (\log_3 5 / \log_{17} 5) = 5 \log_3 17 > 5 \cdot 2 = 10

, so

x > y

Answer:

x > y

(Option C).

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