1. The problem is to solve the equation $$x^x = 5^{x+25}$$ for $x$. 2. Rewrite the right side to separate the exponent, $$5^{x+25} = 5^x \cdot 5^{25}$$. 3. Substitute this back into the equation: $$x^x = 5^x \cdot 5^{25}$$. 4. Divide both sides by $5^x$ to isolate the powers: $$\frac{x^x}{5^x} = 5^{25}$$. 5. Express the left side as $$\left(\frac{x}{5}\right)^x = 5^{25}$$. 6. To solve for $x$, take the natural logarithm of both sides: $$\ln\left(\left(\frac{x}{5}\right)^x\right) = \ln(5^{25})$$. 7. Using logarithm properties, $$x \ln\left(\frac{x}{5}\right) = 25 \ln(5)$$. 8. Let $$y = \frac{x}{5}$$, then $$x = 5y$$, so substituting gives $$5y \ln(y) = 25 \ln(5)$$. 9. Divide both sides by 5: $$y \ln(y) = 5 \ln(5)$$. 10. We want to find $y$ such that $$y \ln(y) = 5 \ln(5)$$. 11. Note that when $y=5$, $$5 \ln(5) = 5 \ln(5)$$ which matches exactly. 12. Therefore, $$y=5$$, so $$x = 5y = 5 \times 5 = 25$$. Final answer: $$\boxed{25}$$.