Test Index

UPSC CDS 2 2021 Math Paper

Question : 20 of 100

Marks: +1, -0

5^{x^{-1}}

(2.5)^{\log_{10}5}

x

Solution:

Concept:

Apply logarithms to solve an exponential equation by converting exponents into multiplicative factors.

Explanation:

Start with the given equation:

5^{x-1} = (2.5)^{\log_{10}5}

Take

\log_{10}

on both sides:

\log_{10}(5^{x-1}) = \log_{10}\left((2.5)^{\log_{10}5}\right)

Use the power rule of logs:

(x-1) \log_{10}5 = (\log_{10}5) \cdot \log_{10}2.5

Since

\log_{10}5 \neq 0

, divide both sides by

\log_{10}5

x-1 = \log_{10}2.5

Add 1 to both sides:

x = 1 + \log_{10}2.5

Write 1 as

\log_{10}10

x = \log_{10}10 + \log_{10}2.5

Use the product rule:

x = \log_{10}(10 \times 2.5) = \log_{10}25

Since

25 = 5^2

x = \log_{10}5^2 = 2\log_{10}5

Answer:

x = 2\log_{10}5

(Option D).

Go to Question:

Prev Question

Next Question