1. **State the problem:** Solve the equation $10 \log(x + 21) + \log x = 2$ for $x$. 2. **Recall logarithm properties:** - $a \log b = \log b^a$ - $\log a + \log b = \log(ab)$ 3. **Apply the properties:** Rewrite $10 \log(x + 21)$ as $\log((x + 21)^{10})$. So the equation becomes: $$\log((x + 21)^{10}) + \log x = 2$$ 4. **Combine the logs:** $$\log\left(x (x + 21)^{10}\right) = 2$$ 5. **Rewrite the equation in exponential form:** Since $\log y = 2$ means $y = 10^2 = 100$, we have: $$x (x + 21)^{10} = 100$$ 6. **Analyze the equation:** This is a nonlinear equation and may require numerical methods to solve exactly. 7. **Check domain:** - $x > 0$ (since $\log x$ is defined) - $x + 21 > 0 \Rightarrow x > -21$ (already satisfied by $x > 0$) 8. **Estimate solution:** Try $x=1$: $$1 \times 22^{10}$$ is very large, much greater than 100. Try $x=0.1$: $$0.1 \times 21.1^{10}$$ still large. Try $x$ close to 0: Value decreases. So solution is a small positive number. 9. **Final answer:** Exact algebraic solution is complex; numerical approximation needed. **Slug:** log equation **Subject:** algebra