We are tasked with simplifying and evaluating the expression:

$\log_3(15) - \log_3(35) + \log_3(189)$

Step 1: Apply the Laws of Logarithms

We can use the following logarithmic properties:

1. Difference of logs:
   $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$

2. Sum of logs:
   $\log_b(x) + \log_b(y) = \log_b(xy)$

Now, let's first combine the terms $\log_3(15) - \log_3(35)$:

$\log_3(15) - \log_3(35) = \log_3\left(\frac{15}{35}\right)$

We can simplify the fraction $\frac{15}{35}$:

$\frac{15}{35} = \frac{3}{7}$

So the expression becomes:

$\log_3\left(\frac{3}{7}\right) + \log_3(189)$

Step 2: Apply the Sum of Logs Rule

Now, use the sum rule to combine the two logarithms:

$\log_3\left(\frac{3}{7}\right) + \log_3(189) = \log_3\left(\frac{3}{7} \times 189\right)$

Step 3: Simplify the Product

Next, calculate the product:

$\frac{3}{7} \times 189 = \frac{3 \times 189}{7} = \frac{567}{7} = 81$

Thus, the expression simplifies to:

$\log_3(81)$

Step 4: Evaluate the Logarithm

Finally, since $81 = 3^4$, we have:

$\log_3(81) = \log_3(3^4) = 4$

Final Answer:

$\log_3(15) - \log_3(35) + \log_3(189) = 4$

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Would you like further clarification or any details? Here are some related questions you may find interesting:

1. How do the properties of logarithms help to simplify expressions like this?
2. What happens if the base of the logarithms changes, for example to base 10 or base $e$?
3. How can we apply the laws of logarithms to solve equations involving logarithms?
4. What is the inverse of a logarithmic function, and how does it relate to exponential functions?
5. Can we apply the logarithmic laws to expressions with different bases (e.g., $\log_2(8)$)?

Tip: When simplifying logarithmic expressions, always look for opportunities to use the product, quotient, and power rules.