If 5^{x-1}=(2.5)^{₁0}5}, then what is the value of x ?

Correct Answer is : 2log⁡1052₁0}52log105

Correct Answer is : 2log⁡1052₁0}52log105

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UPSC CDS 2 2021 Math Paper

Question : 65 of 100

Marks: +1, -0

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

x

Solution: 👈: Video Solution

Given,

5\log_{10}^{(25)}=(2.5)\log_{10}5

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

[\because a^{\log(b)}=b^{\log(a)}]

On comparing LHS and RHS, we get

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

x-1=\log_{10}\left(\frac{25}{10}\right)

x-1=\log_{10}25-\log_{10}10

\left[\because \log\left(\frac{m}{n}\right)=\log m-\log n\right]

x-1=\log_{10}5^{2}-1\;\;[\because \log_{a}a=1]

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

x=2\log_{10}5

[\because \log a^{m}=m\log a]

Hence, the value of

x

2\log_{10}5

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