1. **State the problem:** Solve the equation $5^{2x} - 4 = 3(5^x)$ for $x$. 2. **Rewrite the equation:** Recall that $5^{2x} = (5^x)^2$. Let $y = 5^x$. Then the equation becomes: $$y^2 - 4 = 3y$$ 3. **Rearrange the equation:** Move all terms to one side: $$y^2 - 3y - 4 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=-3$, and $c=-4$. 5. **Calculate the discriminant:** $$\Delta = (-3)^2 - 4(1)(-4) = 9 + 16 = 25$$ 6. **Find the roots:** $$y = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2}$$ 7. **Evaluate each root:** - For $+$ sign: $$y = \frac{3 + 5}{2} = \frac{8}{2} = 4$$ - For $-$ sign: $$y = \frac{3 - 5}{2} = \frac{-2}{2} = -1$$ 8. **Check for valid solutions:** Since $y = 5^x$ and $5^x > 0$ for all real $x$, $y = -1$ is invalid. 9. **Solve for $x$ using valid root:** $$5^x = 4$$ Take logarithm base 5 on both sides: $$x = \log_5 4 = \frac{\ln 4}{\ln 5}$$ **Final answer:** $$x = \frac{\ln 4}{\ln 5}$$