https://file.aiquickdraw.com/mathbot/file/1729480078506o4ktqps9.jpg

Let's analyze the problem shown in the image:

### **Given:**
\[
\log_7(11 \cdot 5)
\]

### **Goal:**
Use properties of logarithms to expand the expression.

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### **Solution:**
One of the key properties of logarithms is the **product rule**, which states:
\[
\log_b(A \cdot B) = \log_b(A) + \log_b(B)
\]
So, applying this rule to the given expression:

\[
\log_7(11 \cdot 5) = \log_7(11) + \log_7(5)
\]

### **Final Expanded Expression:**
\[
\log_7(11) + \log_7(5)
\]

Since \( \log_7(11) \) and \( \log_7(5) \) are not perfect powers of 7, we leave them in this form. This is the most expanded version possible without a calculator.

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Do you have any further questions or need more details?  

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Here are five related questions for deeper exploration:  
1. How do you apply the quotient rule for logarithms?  
2. What happens if you have a logarithm with a power, like \( \log_b(A^n) \)?  
3. How do you evaluate logarithms with different bases (e.g., base 2 or base 10)?  
4. What is the change of base formula, and when is it used?  
5. How can logarithmic properties help solve equations involving exponential terms?

**Tip:** Always look for product or quotient terms inside a logarithm to use the product or quotient rules effectively.

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7(11 * 5)

Learn how to expand the logarithmic expression log7(11 * 5) using the product rule for logarithms. This problem focuses on key algebraic concepts, including logarithmic properties and their application. Suitable for high school students in Grades 9-12.

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