1. **State the problem:** Solve the equation $5^{2x} - 4 = 3(5^x)$. 2. **Rewrite the equation:** Recall that $5^{2x} = (5^x)^2$. Let $y = 5^x$. 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:** $$y = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2}$$ So, $y_1 = \frac{3 + 5}{2} = 4$ and $y_2 = \frac{3 - 5}{2} = -1$. 7. **Back-substitute for $y$:** Recall $y = 5^x$. Since $5^x > 0$ for all real $x$, $y = -1$ is not valid. 8. **Solve for $x$:** $5^x = 4$. Taking logarithm base 5, $$x = \log_5 4 = \frac{\ln 4}{\ln 5}$$ 9. **Final answer:** $$x = \log_5 4$$