https://file.aiquickdraw.com/mathbot/file/1731032883583f1569ian.jpg

To expand the given logarithmic expression, let's use logarithmic rules step-by-step. The expression we need to expand is:

\[
\ln \left( \left( \frac{a^{-2} b^4 c^7}{a^7 b^2 c^{-5}} \right)^2 \cdot \left( a^4 b^{-13} c^6 \right)^2 \right)
\]

### Step-by-Step Solution:

1. **Apply the Power Rule for Logarithms:** The entire expression is raised to the power of \(2\), so we can bring this \(2\) out in front of the logarithm.

   \[
   = 2 \cdot \ln \left( \frac{a^{-2} b^4 c^7}{a^7 b^2 c^{-5}} \cdot a^4 b^{-13} c^6 \right)
   \]

2. **Separate the Product using the Sum Rule for Logarithms:** Since there is multiplication inside the logarithm, we can separate it as a sum of logs.

   \[
   = 2 \cdot \left( \ln \left( \frac{a^{-2} b^4 c^7}{a^7 b^2 c^{-5}} \right) + \ln \left( a^4 b^{-13} c^6 \right) \right)
   \]

3. **Apply the Quotient Rule for Logarithms in the First Term:** Now, for the first term inside the log, we have a division. We can separate this into a subtraction of logs.

   \[
   = 2 \cdot \left( \left( \ln (a^{-2} b^4 c^7) - \ln (a^7 b^2 c^{-5}) \right) + \ln (a^4 b^{-13} c^6) \right)
   \]

4. **Expand Each Logarithmic Expression Using the Power Rule:** Now, expand each term inside the parentheses by breaking them down into individual logs and bringing the exponents in front.

   \[
   = 2 \cdot \left( \left( -2 \ln a + 4 \ln b + 7 \ln c \right) - \left( 7 \ln a + 2 \ln b - 5 \ln c \right) + \left( 4 \ln a - 13 \ln b + 6 \ln c \right) \right)
   \]

5. **Distribute and Simplify:** Now, expand and combine like terms.

   - First, distribute the negative sign in the second set of terms:
   
     \[
     = 2 \cdot \left( -2 \ln a + 4 \ln b + 7 \ln c - 7 \ln a - 2 \ln b + 5 \ln c + 4 \ln a - 13 \ln b + 6 \ln c \right)
     \]

   - Combine the terms for \( \ln a \), \( \ln b \), and \( \ln c \) separately:

     - For \( \ln a \): \( -2 - 7 + 4 = -5 \)
     - For \( \ln b \): \( 4 - 2 - 13 = -11 \)
     - For \( \ln c \): \( 7 + 5 + 6 = 18 \)

     So we have:

     \[
     = 2 \cdot \left( -5 \ln a - 11 \ln b + 18 \ln c \right)
     \]

6. **Final Distribution:** Multiply each term by \(2\) to get the final expanded form.

   \[
   = -10 \ln a - 22 \ln b + 36 \ln c
   \]

### Final Answer:

\[
\ln \left( \left( \frac{a^{-2} b^4 c^7}{a^7 b^2 c^{-5}} \right)^2 \cdot \left( a^4 b^{-13} c^6 \right)^2 \right) = -10 \ln a - 22 \ln b + 36 \ln c
\]

Would you like further details on any of the steps?

---

Here are some related questions for further exploration:

1. How do logarithmic properties help simplify complex expressions in general?
2. What is the difference between natural logarithms and common logarithms?
3. How does the Power Rule for logarithms apply in other mathematical contexts?
4. Can logarithmic functions be used to solve exponential equations?
5. How does expanding logarithmic expressions help in calculus, particularly in differentiation?

**Tip:** Always remember that logarithmic identities are essential for simplifying expressions, especially in calculus and algebra, where complex expressions often appear.

Use log rules to expand the following log completely: \( \ln \left( \left( \frac{a^{-2} b^4 c^7}{a^7 b^2 c^{-5}} \right)^2 \cdot \left( a^4 b^{-13} c^6 \right)^2 \right) \)

Learn how to expand the logarithmic expression \( \ln \left( \left( \frac{a^{-2} b^4 c^7}{a^7 b^2 c^{-5}} \right)^2 \cdot \left( a^4 b^{-13} c^6 \right)^2 \right) \) using log rules like the power, product, and quotient rules. This step-by-step solution covers key concepts for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation