1. **State the problem:** Simplify the expression $$\frac{\log_5\left(10 \cdot \left(1000^{\frac{1}{2}}\right)\right)}{\log_5(0.01)}$$. 2. **Recall the logarithm properties:** - Product rule: $$\log_b(xy) = \log_b x + \log_b y$$ - Power rule: $$\log_b(x^r) = r \log_b x$$ - Change of base is not needed here since the base is consistent. 3. **Simplify the numerator:** $$\log_5\left(10 \cdot 1000^{\frac{1}{2}}\right) = \log_5 10 + \log_5 \left(1000^{\frac{1}{2}}\right)$$ 4. **Simplify the power term:** $$\log_5 \left(1000^{\frac{1}{2}}\right) = \frac{1}{2} \log_5 1000$$ 5. **Express 10 and 1000 in terms of powers of 10:** $$10 = 10^1$$ $$1000 = 10^3$$ 6. **Rewrite logs in terms of base 5:** $$\log_5 10 = \log_5 \left(5 \cdot 2\right) = \log_5 5 + \log_5 2 = 1 + \log_5 2$$ $$\log_5 1000 = \log_5 10^3 = 3 \log_5 10 = 3(1 + \log_5 2)$$ 7. **Substitute back:** $$\log_5 10 + \frac{1}{2} \log_5 1000 = (1 + \log_5 2) + \frac{1}{2} \times 3(1 + \log_5 2) = (1 + \log_5 2) + \frac{3}{2}(1 + \log_5 2)$$ 8. **Combine terms:** $$= (1 + \log_5 2) + 1.5(1 + \log_5 2) = (1 + \log_5 2)(1 + 1.5) = (1 + \log_5 2)(2.5)$$ 9. **Simplify the denominator:** $$\log_5(0.01) = \log_5 \left(\frac{1}{100}\right) = \log_5 \left(10^{-2}\right) = -2 \log_5 10 = -2(1 + \log_5 2)$$ 10. **Form the fraction:** $$\frac{(1 + \log_5 2)(2.5)}{-2(1 + \log_5 2)}$$ 11. **Cancel common factor:** $$= \frac{\cancel{(1 + \log_5 2)} 2.5}{-2 \cancel{(1 + \log_5 2)}} = \frac{2.5}{-2} = -\frac{5}{4}$$ **Final answer:** $$-\frac{5}{4}$$