To evaluate the even-numbered logarithmic expressions from the provided exercises, I'll follow the steps laid out using logarithmic properties, and where necessary, approximate values from the provided table.

Let's begin:

2. $5\log_5 45 - \log_5 9$

Using logarithmic properties: $\log_b(mn) = \log_b m + \log_b n \quad \text{and} \quad \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n$

This simplifies as follows: $5\log_5 45 = \log_5 45^5 \quad \text{and} \quad \log_5 9 = \log_5 9$ From the table, approximate values can be used: $\log_5 45 \approx 2.6561 \quad \text{and} \quad \log_5 9 \approx 1.4307$ Thus, $5(2.6561) - 1.4307 = 13.2805 - 1.4307 = 11.8498$

4. $\log_5 30 - \log_3 4$

Using properties of logarithms: $\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n$ This expression becomes: $\log_5 \left(\frac{30}{4}\right)$ We can approximate the values based on the table: $\log_5 30 \approx 2.6826 \quad \text{and} \quad \log_3 4 \approx 1.2619$ Thus, $2.6826 - 1.2619 = 1.4207$

6. $\log_6 1 + \log_6 36 + \log_6 5$

Since $\log_b 1 = 0$, the expression simplifies to: $0 + \log_6 36 + \log_6 5 = \log_6 180$ From the logarithmic properties or table, we approximate: $\log_6 180 \approx 2.7284$

8. $\log_3 209$

From the table, approximate: $\log_3 209 \approx 4.5282$

10. $\log_2 6 + \log_2 8 + \log_2 16$

Using the property $\log_b(mn) = \log_b m + \log_b n$, this becomes: $\log_2(6 \times 8 \times 16) = \log_2 768$ From the table, approximate: $\log_2 768 \approx 9.589$

These are the evaluated even-numbered expressions. Do you need further clarification on any step?

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Related Questions:

1. What are the logarithmic properties used to combine or separate logarithmic terms?
2. How would you simplify expressions involving both addition and subtraction of logarithms?
3. How is the change-of-base formula used to calculate logarithms with a base not in the table?
4. What is the significance of $\log_b 1 = 0$?
5. How can exponential functions help simplify complex logarithmic expressions?

Tip: When approximating logarithms, using properties such as $\log_b \left( m \times n \right)$ can greatly simplify calculations.