If $ 25^{x-1}=5^{2 x-1}-100 $, find the value of $ x $.

Given:

\( 25^{x-1}=5^{2 x-1}-100 \)

To do:

We have to find the value of $x$.

Solution:

$25^{x-1}=5^{2x-1}-100$

$(5^2)^{x-1}=5^{2x-1}-100$

$5^{2x-2}-5^{2x-1}=-100$

$5^{2x-1}(5^{-1}-1)=-100$

$5^{2x-1}(\frac{1}{5}-1)=-100$

$5^{2x-1}(1-\frac{1}{5})=100$

$5^{2x-1}(\frac{5-1}{5})=100$

$5^{2x-1}(\frac{4}{5})=100$

$5^{2x-1}(2^2)\times5^{-1}=(25\times4)$

$5^{2x-2}(2^2)=(5^2\times2^2)$

Comparing both sides, we get,

$2x-2=2$

$2x=2+2$

$2x=4$

$x=2$

Therefore, the value of $x$ is $2$. 

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Simply Easy Learning

Updated on: 10-Oct-2022

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