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The logarithmic expression you have is:

\[
\log_6(7 \cdot 11)
\]

Using the properties of logarithms, particularly the product rule, which states:

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

we can expand the expression as:

\[
\log_6(7 \cdot 11) = \log_6(7) + \log_6(11)
\]

This is the expanded form. Since there's no indication that \(\log_6(7)\) or \(\log_6(11)\) can be simplified further without a calculator, this is the final expanded form.

Would you like any further details or explanations?

Here are five related questions:
1. How do you expand a logarithmic expression involving division?
2. What is the change of base formula for logarithms?
3. Can you simplify the sum \(\log_6(7) + \log_6(11)\) using any approximation or numeric methods?
4. How would the expression change if it were \(\log_6(77)\) instead of \(\log_6(7 \cdot 11)\)?
5. How does the logarithmic power rule apply to terms with exponents?

**Tip:** Always check whether the arguments of a logarithmic function can be factored or simplified before applying rules.

Use properties of logarithms to expand the logarithmic expression log_6(7 * 11). Where possible, evaluate logarithmic expressions without using a calculator.

Learn how to expand the logarithmic expression log_6(7 * 11) using the product rule. This problem covers key logarithmic properties and shows step-by-step how to apply them for proper expansion. Suitable for high school students in Grades 10-12.

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