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UPSC CDS I 2025 Mathematics Paper

Question : 20 of 100

Marks: +1, -0

\;\frac{\log_{10}(100001-4^{x})}{5-x} = 1

, then what is x equal

Solution:

Given:

\;\frac{\log_{10}(100001-4^{x})}{5-x}=1

Formula used:

\log a = b \Rightarrow a = 10^{b}

Calculation:

\;\Rightarrow \log_{10}(100001-4^{x}) = 5-x

\;\Rightarrow 100001-4^{x}=10^{5-x}

\;\Rightarrow 100001-4^{x}=10^{5-x}

Check all the options:

\;x = 0

\;\Rightarrow 100001-4^{0}=10^{5-0}

\;\Rightarrow 100001-1=100000

x = 0

satisfies the condition.

\;x = 1

\;\Rightarrow 100001-4^{1}=10^{5-1}

\;\Rightarrow 100001-4=10000

\;\Rightarrow 99997 \neq 10000

x = 1

does not satisfy the condition.

\;x = 1 0

\;\Rightarrow 100001-4^{10}=10^{5-10}

\;\Rightarrow 100001-4^{10} \neq 10^{-5}

x=10

does not satisfy the condition.

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