Question

Find $x$, if $5^{2x-1}-{25}^{x-1}=2500$.

• A. $-3$
• B. $5$
• C. $3$
• D. $-1$

Answer 1

The correct option is C

$3$

Explanation for correct option:

Solve the equation for $x$:

Given, $5^{2x-1}-{25}^{x-1}=2500$

$\begin{array}{cc}\Rightarrow & 5^{2x-1}-{\left(5\right)}^{2x-2}=2500\\ \Rightarrow & 5^{2x-1}-\frac{{\left(5\right)}^{2x-1}}{5}=2500\end{array}$

Let $5^{2x-1}$ be equal to $a$

$\begin{array}{cc}\Rightarrow & a-\frac{a}{5}=2500\\ \Rightarrow & \frac{5a-a}{5}=2500\\ \Rightarrow & \frac{4a}{5}=2500\\ \Rightarrow & a=\frac{2500\times 4}{5}\\ \Rightarrow & a=3125\end{array}$

Now as we know that $5^{2x-1}=a$

So, $5^{2x-1}=3125$

$$\begin{gathered} \Rightarrow 5^{2x-1}=5^5\left[\because a^m=a^n\right] \\ \Rightarrow 2x-1=5\left[\because m=n\right] \\ \Rightarrow 2x=5+1 \\ \Rightarrow x=\frac{6}{2} \\ \Rightarrow x=3 \end{gathered}$$

Therefore, $x$ is equal to $3$.