Question

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

Answer 1

Find the value of $x$.

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

$\Rightarrow 5^{2\left(x-1\right)}-5^{2x-1}=-100\left[\because 25=5^2\right]$

$$\begin{gathered} \Rightarrow 5^{2x-1}-5^{2x-2}=100 \\ \Rightarrow 5^{2x-2}\left(5^1-1\right)=100 \\ \Rightarrow 5^{2(x-1)}=\frac{100}{4} \\ \Rightarrow {25}^{(x-1)}=25 \end{gathered}$$

Comparing the powers in both the sides:

$$\begin{gathered} x-1=1 \\ \Rightarrow \boxed{x=2} \end{gathered}$$

Hence, the value of $x$ is $2$.