Question

Please help me out with this problem.
$25 { }^{x - 1} = 5 {}^{2x - 1} - 100$
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Answer 1

$\begin{aligned}&25^{x - 1} = 5^{2x - 1} - 100\\ \\ \Longrightarrow\ \ &5^{2x-1}-25^{x-1}=100\\ \\ \Longrightarrow\ \ &5^{2x-1}-(5^2)^{x-1}=100\\ \\ \Longrightarrow\ \ &5^{2x-1}-5^{2x-2}=100\\ \\ \Longrightarrow\ \ &5^{2x-2+1}-5^{2x-2}=100\\ \\ \Longrightarrow\ \ &5^{2x-2}(5-1)=100\\ \\ \Longrightarrow\ \ &25^{x-1}\times 4=100\end{aligned}$

$\begin{aligned}\Longrightarrow\ \ &25^{x-1}=25\\ \\ \Longrightarrow\ \ &x-1=1\\ \\ \Longrightarrow\ \ &x=\mathbf{2}\end{aligned}$

Hence the answer is 2.

Let's check.

$\begin{aligned}&\large\textsf{LHS...}\\ \\ \Longrightarrow\ \ &25^{x-1}\\ \\ \Longrightarrow\ \ &25^{2-1}\\ \\ \Longrightarrow\ \ &25\\ \\ \Longrightarrow\ \ &25+100-100\\ \\ \Longrightarrow\ \ &125-100\\ \\ \Longrightarrow\ \ &5^3-100\\ \\ \Longrightarrow\ \ &5^{4-1}-100\\ \\ \Longrightarrow\ \ &5^{2\times 2-1}-100\\ \\ \Longrightarrow\ \ &5^{2x-1}-100\end{aligned}$

Hence checked!