10) \( \log \left(2^{2} \cdot 3\right)^{6} \) 12) \( \log _{4}\left(x \cdot y \cdot z^{4}\right) \) 14) \( 5 \log _{3} a+25 \log _{3} b \) 20) \( 5 \log _{8} 5-30 \log _{8} 12 \) 18) \( 10 \log _{9} a+5 \log _{9} b \) 16) \( 3 \log _{3} a-18 \log _{3} b \)

Sure, let's simplify each of the given logarithmic expressions using logarithm properties such as the **Product Rule**, **Quotient Rule**, and **Power Rule**. ### 10) \( \log \left(2^{2} \cdot 3\right)^{6} \) **Step-by-Step Simplification:** 1. **Apply the Power Rule:** \(\log (A^k) = k \log A\) \[ \log \left(2^{2} \cdot 3\right)^{6} = 6 \log (2^{2} \cdot 3) \] 2. **Use the Product Rule:** \(\log (AB) = \log A + \log B\) \[ 6 \left(\log 2^{2} + \log 3\right) \] 3. **Apply the Power Rule Again:** \[ 6 \left(2 \log 2 + \log 3\right) = 12 \log 2 + 6 \log 3 \] **Final Simplified Form:** \[ \boxed{12\,\log 2\,+\,6\,\log 3} \] --- ### 12) \( \log _{4}\left(x \cdot y \cdot z^{4}\right) \) **Step-by-Step Simplification:** 1. **Use the Product Rule:** \[ \log_4 (x) + \log_4 (y) + \log_4 (z^{4}) \] 2. **Apply the Power Rule:** \[ \log_4 x + \log_4 y + 4 \log_4 z \] **Final Simplified Form:** \[ \boxed{\log_4 x\,+\,\log_4 y\,+\,4\,\log_4 z} \] --- ### 14) \( 5 \log _{3} a + 25 \log _{3} b \) **Step-by-Step Simplification:** 1. **Apply the Power Rule to each term:** \[ \log_3 a^{5} + \log_3 b^{25} \] 2. **Use the Product Rule:** \[ \log_3 (a^{5} \cdot b^{25}) \] **Final Simplified Form:** \[ \boxed{\log_3 (a^{5}b^{25})} \] --- ### 20) \( 5 \log _{8} 5 - 30 \log _{8} 12 \) **Step-by-Step Simplification:** 1. **Apply the Power Rule to each term:** \[ \log_8 5^{5} - \log_8 12^{30} \] 2. **Use the Quotient Rule:** \(\log_b A - \log_b C = \log_b \left(\frac{A}{C}\right)\) \[ \log_8 \left( \frac{5^{5}}{12^{30}} \right) \] **Final Simplified Form:** \[ \boxed{\log_8 \left(\dfrac{5^{5}}{12^{30}}\right)} \] --- ### 18) \( 10 \log _{9} a + 5 \log _{9} b \) **Step-by-Step Simplification:** 1. **Apply the Power Rule to each term:** \[ \log_9 a^{10} + \log_9 b^{5} \] 2. **Use the Product Rule:** \[ \log_9 (a^{10} \cdot b^{5}) \] **Final Simplified Form:** \[ \boxed{\log_9 (a^{10}b^{5})} \] --- ### 16) \( 3 \log _{3} a - 18 \log _{3} b \) **Step-by-Step Simplification:** 1. **Apply the Power Rule to each term:** \[ \log_3 a^{3} - \log_3 b^{18} \] 2. **Use the Quotient Rule:** \[ \log_3 \left( \frac{a^{3}}{b^{18}} \right) \] **Final Simplified Form:** \[ \boxed{\log_3 \left(\dfrac{a^{3}}{b^{18}}\right)} \] --- Each expression has been simplified using fundamental logarithmic properties. If you have any further questions or need additional explanations, feel free to ask!