Answer
Here are the simplified forms of the given logarithmic expressions:
1. \( \log \left(2^{2} \cdot 3\right)^{6} = 12\,\log 2\,+\,6\,\log 3 \)
2. \( \log _{4}\left(x \cdot y \cdot z^{4}\right) = \log_4 x\,+\,\log_4 y\,+\,4\,\log_4 z \)
3. \( 5 \log _{3} a + 25 \log _{3} b = \log_3 (a^{5}b^{25}) \)
4. \( 5 \log _{8} 5 - 30 \log _{8} 12 = \log_8 \left(\dfrac{5^{5}}{12^{30}}\right) \)
5. \( 10 \log _{9} a + 5 \log _{9} b = \log_9 (a^{10}b^{5}) \)
6. \( 3 \log _{3} a - 18 \log _{3} b = \log_3 \left(\dfrac{a^{3}}{b^{18}}\right) \)
Solution
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!
Reviewed and approved by the UpStudy tutoring team
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