1. **Problem:** Expand each logarithm using logarithm properties. 2. **Step 1:** $\log_6 \left(\frac{u}{v}\right)$ - Use the quotient rule: $\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$ - So, $\log_6 \left(\frac{u}{v}\right) = \log_6 u - \log_6 v$ 3. **Step 2:** $\log_5 \sqrt[3]{a}$ - Use the power rule: $\log_b (M^p) = p \log_b M$ - $\sqrt[3]{a} = a^{\frac{1}{3}}$ - So, $\log_5 \sqrt[3]{a} = \frac{1}{3} \log_5 a$ 4. **Step 3:** $\log_7 5^4$ - Use the power rule - $\log_7 5^4 = 4 \log_7 5$ 5. **Step 4:** $\log_4 4^6$ - Use the power rule - $\log_4 4^6 = 6 \log_4 4$ - Since $\log_4 4 = 1$, this simplifies to $6$ 6. **Step 5:** $\log (a \cdot b)$ - Use the product rule: $\log_b (MN) = \log_b M + \log_b N$ - So, $\log (a \cdot b) = \log a + \log b$ 7. **Step 6:** $\log_5 \left(\frac{6}{7}\right)$ - Use the quotient rule - $\log_5 \left(\frac{6}{7}\right) = \log_5 6 - \log_5 7$ --- 8. **Problem:** Condense each expression to a single logarithm. 9. **Step 7:** $6 \log_5 10$ - Use power rule in reverse: $a \log_b M = \log_b M^a$ - So, $6 \log_5 10 = \log_5 10^6$ 10. **Step 8:** $\frac{\log x}{3}$ - Rewrite as $\frac{1}{3} \log x$ - Use power rule in reverse - $\frac{\log x}{3} = \log x^{\frac{1}{3}} = \log \sqrt[3]{x}$ 11. **Step 9:** $\log_7 u - \log_7 v$ - Use quotient rule in reverse - $\log_7 u - \log_7 v = \log_7 \left(\frac{u}{v}\right)$ 12. **Step 10:** $\log_6 x - \log_6 y$ - Use quotient rule in reverse - $\log_6 x - \log_6 y = \log_6 \left(\frac{x}{y}\right)$ 13. **Step 11:** $\log_4 2 + \log_4 7$ - Use product rule in reverse - $\log_4 2 + \log_4 7 = \log_4 (2 \cdot 7) = \log_4 14$ 14. **Step 12:** $\log_3 a + \log_3 b$ - Use product rule in reverse - $\log_3 a + \log_3 b = \log_3 (ab)$ **Final answers:** - Expanded: 1) $\log_6 u - \log_6 v$ 2) $\frac{1}{3} \log_5 a$ 3) $4 \log_7 5$ 4) $6$ 5) $\log a + \log b$ 6) $\log_5 6 - \log_5 7$ - Condensed: 19) $\log_5 10^6$ 20) $\log \sqrt[3]{x}$ 21) $\log_7 \left(\frac{u}{v}\right)$ 22) $\log_6 \left(\frac{x}{y}\right)$ 23) $\log_4 14$ 24) $\log_3 (ab)$