1. **State the problem:** Solve the equation $$4 + \log\left(\frac{x}{4}\right) = \log(x^2)$$ for $x$. 2. **Recall logarithm properties:** - $\log(a) + \log(b) = \log(ab)$ - $\log(a^b) = b\log(a)$ - The equation involves logarithms with the same base (assumed base 10). 3. **Isolate logarithmic terms:** Rewrite the equation as $$4 + \log\left(\frac{x}{4}\right) = \log(x^2)$$ 4. **Express the constant 4 as a logarithm:** Since $4 = \log(10^4)$, rewrite: $$\log(10^4) + \log\left(\frac{x}{4}\right) = \log(x^2)$$ 5. **Combine the left side logarithms:** $$\log\left(10^4 \times \frac{x}{4}\right) = \log(x^2)$$ 6. **Simplify inside the logarithm:** $$\log\left(\frac{10^4 x}{4}\right) = \log(x^2)$$ 7. **Since $\log(a) = \log(b)$ implies $a = b$, set arguments equal:** $$\frac{10^4 x}{4} = x^2$$ 8. **Multiply both sides by 4 to clear denominator:** $$10^4 x = 4 x^2$$ 9. **Rewrite and simplify:** $$10^4 x - 4 x^2 = 0$$ 10. **Factor out $x$:** $$x(10^4 - 4 x) = 0$$ 11. **Set each factor equal to zero:** - $x = 0$ (not valid since $\log(0)$ is undefined) - $10^4 - 4 x = 0$ 12. **Solve for $x$:** $$4 x = 10^4$$ $$x = \frac{10^4}{4}$$ 13. **Calculate the value:** $$x = \frac{10000}{4} = 2500$$ 14. **Check domain:** $x=2500 > 0$, valid for logarithms. **Final answer:** $$\boxed{2500}$$