The given expression is:

$\log_3 72 - \log_3 8 + \log_3 \left(\frac{1}{3}\right) + \log_3 48 - \log_3 16$

We can simplify this step-by-step using the properties of logarithms. Recall the following rules:

1. $\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)$
2. $\log_b a + \log_b c = \log_b (a \cdot c)$

Step 1: Combine the terms

Using the subtraction and addition rules for logarithms, combine the terms:

$\log_3 \left(\frac{72}{8}\right) + \log_3 \left(\frac{1}{3} \cdot 48 \cdot \frac{1}{16}\right)$

Step 2: Simplify the fractions

• $\frac{72}{8} = 9$
• Now, $\frac{1}{3} \cdot 48 \cdot \frac{1}{16} = \frac{48}{48} = 1$

So the expression becomes:

$\log_3 9 + \log_3 1$

Step 3: Evaluate the logarithms

• $\log_3 9 = 2$ (since $9 = 3^2$)
• $\log_3 1 = 0$ (since $\log_b 1 = 0$ for any base $b$)

Final Simplified Expression:

$2 + 0 = 2$

Thus, the value of the expression is 2.

The correct answer is e. 2.

Would you like any more details or further clarification?

Here are 5 related questions for further exploration:

1. How can you simplify logarithmic expressions involving multiple logarithms?
2. What are the key properties of logarithms used in solving such equations?
3. Can logarithmic expressions be solved without converting to exponentials?
4. How do logarithmic bases affect the final value of a logarithmic expression?
5. What is the relationship between exponents and logarithms in solving equations?

Tip: When solving logarithmic equations, always look for opportunities to combine or simplify using logarithmic rules such as addition, subtraction, or converting to exponents.