https://file.aiquickdraw.com/mathbot/file/17285094316969skoudbx.jpg

The given expression is:

\[
\log_2 \left( \frac{6x^4}{y^7} \right)
\]

We are asked to expand it using the properties of logarithms. Specifically, we will use these rules:

1. **Product Rule**: \(\log_b(mn) = \log_b(m) + \log_b(n)\)
2. **Quotient Rule**: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)
3. **Power Rule**: \(\log_b(m^n) = n \cdot \log_b(m)\)

### Step-by-step Expansion:

1. **Apply the Quotient Rule**:
   \[
   \log_2 \left( \frac{6x^4}{y^7} \right) = \log_2(6x^4) - \log_2(y^7)
   \]

2. **Apply the Product Rule** to \(\log_2(6x^4)\):
   \[
   \log_2(6x^4) = \log_2(6) + \log_2(x^4)
   \]

3. **Apply the Power Rule** to \(\log_2(x^4)\) and \(\log_2(y^7)\):
   \[
   \log_2(x^4) = 4 \log_2(x) \quad \text{and} \quad \log_2(y^7) = 7 \log_2(y)
   \]

### Final Expanded Expression:

\[
\log_2 \left( \frac{6x^4}{y^7} \right) = \log_2(6) + 4 \log_2(x) - 7 \log_2(y)
\]

So, to fill in the blanks from the image:

\[
\log_2(?) + \log_2(?) - \log_2(?)
\]

should be:

\[
\log_2(6) + 4 \log_2(x) - 7 \log_2(y)
\]

Would you like further explanation on any part of this solution? Here are 5 follow-up questions that could help expand your understanding:

1. How is the quotient rule applied to logarithms in general?
2. Why does the power of a variable become a coefficient when expanding logarithms?
3. Can the base of the logarithm change the result of the expansion?
4. What other properties of logarithms might be useful in different types of expressions?
5. How would the expansion change if the numerator or denominator contained more factors?

**Tip**: Always remember to break complex logarithmic expressions into simpler parts using the product, quotient, and power rules!

Learn how to expand the logarithmic expression log_2(6x^4 / y^7) using logarithmic properties like the product, quotient, and power rules. This detailed step-by-step explanation is ideal for high school algebra students.

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