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To simplify the expression \( 2 \log_b 6 - \log_b 2 \) as a single logarithm, let's follow these steps:

1. **Apply the Power Rule:** 
   The term \( 2 \log_b 6 \) can be rewritten using the power rule for logarithms, \( a \log_b x = \log_b (x^a) \).
   \[
   2 \log_b 6 = \log_b (6^2) = \log_b 36
   \]

   So the expression now becomes:
   \[
   \log_b 36 - \log_b 2
   \]

2. **Apply the Quotient Rule:** 
   Now, use the quotient rule for logarithms, which states \( \log_b x - \log_b y = \log_b \left( \frac{x}{y} \right) \).
   \[
   \log_b 36 - \log_b 2 = \log_b \left( \frac{36}{2} \right)
   \]

3. **Simplify the Quotient:**
   \[
   \frac{36}{2} = 18
   \]

   So the expression simplifies to:
   \[
   \log_b 18
   \]

### Final Answer:
\[
2 \log_b 6 - \log_b 2 = \log_b 18
\]

Would you like further details on any of these steps?

Here are some related questions to consider:
1. How do you apply the power rule for logarithms in other contexts?
2. What is the quotient rule for logarithms, and when is it useful?
3. Can you combine different logarithmic expressions with other bases?
4. How do you expand or break down logarithmic expressions with multiple terms?
5. What happens if you add instead of subtract two logarithmic expressions?

**Tip:** Always look for opportunities to apply the power, product, or quotient rules to simplify logarithmic expressions effectively.

Write the expression 2 log_b 6 - log_b 2 as a single logarithm in simplest form.

Learn how to simplify the expression 2 log_b 6 - log_b 2 as a single logarithm. This example covers logarithmic rules, including the power and quotient rules, ideal for high school students in Grades 9-12.

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