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The problem asks to use properties of logarithms to expand and simplify the expression \(\log_3(11 \cdot 7)\).

### Step 1: Apply the Product Rule for Logarithms
The product rule of logarithms states that:

\[
\log_b (x \cdot y) = \log_b x + \log_b y
\]

Here, we apply this rule to the given logarithmic expression:

\[
\log_3 (11 \cdot 7) = \log_3 11 + \log_3 7
\]

Thus, the expanded form is:

\[
\log_3 (11 \cdot 7) = \log_3 11 + \log_3 7
\]

This is the most simplified form using the properties of logarithms. Further numerical evaluation would depend on knowing the exact values of \(\log_3 11\) and \(\log_3 7\), which are not easily calculated without a calculator.

Would you like to explore any further details or calculations? 

### 5 Related Questions:
1. How can the quotient rule of logarithms be applied to logarithmic expressions?
2. What are the steps for expanding logarithmic expressions with powers (like \(\log_b (x^n)\))?
3. Can logarithms with different bases be simplified or converted into the same base?
4. What are the properties of logarithms involving sums and differences?
5. How can we use logarithms to solve exponential equations?

### Tip:
To quickly simplify logarithmic expressions, always remember the key properties: product rule, quotient rule, and power rule for logarithms.

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log_3(11 * 7)

Learn how to expand the logarithmic expression log_3(11 * 7) by applying the product rule of logarithms. This step-by-step guide explores key algebraic concepts and formulas, suitable for students in Grades 10-12.

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