If 5^{x - 3} = 8, then what is x equal to?

Correct Answer is : 31−log⁡102{3}{1-₁0}2}1−log1023

Correct Answer is : 31−log⁡102{3}{1-₁0}2}1−log1023

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UPSC CDS 1 2021 Math Paper

Question : 55 of 100

Marks: +1, -0

5^{x - 3} = 8

x

equal to?

Solution:

Concept:

Using logarithms to solve exponential equations and the property

\log_{10} 5 = 1 - \log_{10} 2

Explanation:

The equation is

5^{x-3} = 8

Take

\log_{10}

on both sides:

(x-3) \log_{10} 5 = \log_{10} 8

Since

8 = 2^3

, we have

\log_{10} 8 = 3 \log_{10} 2

Thus,

x-3 = \frac{3 \log_{10} 2}{\log_{10} 5}

Add 3 to both sides:

x = 3 + \frac{3 \log_{10} 2}{\log_{10} 5} = 3\left(1 + \frac{\log_{10} 2}{\log_{10} 5}\right)

Now note:

\log_{10} 5 = \log_{10} \frac{10}{2} = \log_{10} 10 - \log_{10} 2 = 1 - \log_{10} 2

Substitute:

1 + \frac{\log_{10} 2}{1 - \log_{10} 2} = \frac{1 - \log_{10} 2 + \log_{10} 2}{1 - \log_{10} 2} = \frac{1}{1 - \log_{10} 2}

Therefore,

x = 3 \times \frac{1}{1 - \log_{10} 2} = \frac{3}{1 - \log_{10} 2}

Answer:

Option A:

\frac{3}{1 - \log_{10} 2}

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