Test Index

UPSC CDS 2 2022 Math Paper

Question : 22 of 100

Marks: +1, -0

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

, (a ≠ b ≠ c), then what is the value of abc ?

Solution:

Concept:

Use the property of equal ratios to express each logarithm as a product of a constant

k

and the denominator, then add them to derive

abc = 1

Explanation:

Let the common ratio be

k

Then

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

\log_{10}a = k(b-c)

\log_{10}b = k(c-a)

\log_{10}c = k(a-b)

Add all three equations:

\log_{10}a + \log_{10}b + \log_{10}c = k(b-c + c-a + a-b)

The sum inside the brackets is

0

Thus

\log_{10}a + \log_{10}b + \log_{10}c = 0

Using

\log m + \log n = \log(mn)

, we get

\log_{10}(abc) = 0

Therefore

abc = 10^0 = 1

Answer:

1

Go to Question:

Prev Question

Next Question