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To condense the given expression \(3(\log_3(3) + \log_3(z)) - \log_3(2)\) into a single logarithm, let's go through the steps:

### Step 1: Expand and Simplify the Terms Inside the Parentheses
The expression inside the parentheses is:
\[
\log_3(3) + \log_3(z)
\]
Using the **product rule** for logarithms, which states that \(\log_b(x) + \log_b(y) = \log_b(x \cdot y)\), we can rewrite this as:
\[
\log_3(3 \cdot z) = \log_3(3z)
\]

### Step 2: Apply the Coefficient Outside the Parentheses
Now, multiply this result by 3:
\[
3 \cdot \log_3(3z) = \log_3((3z)^3)
\]
This uses the **power rule** of logarithms, which states that \(a \cdot \log_b(x) = \log_b(x^a)\).

Expanding \((3z)^3\):
\[
(3z)^3 = 27z^3
\]
So, we now have:
\[
\log_3(27z^3)
\]

### Step 3: Combine with the Remaining Term
The expression now reads:
\[
\log_3(27z^3) - \log_3(2)
\]
Using the **quotient rule** for logarithms, which states that \(\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\), we can combine these terms:
\[
\log_3\left(\frac{27z^3}{2}\right)
\]

### Final Answer
The condensed expression is:
\[
\log_3\left(\frac{27z^3}{2}\right)
\]

Would you like further details on any specific step?

### Further Questions
1. What is the power rule for logarithms and how can it be applied in different scenarios?
2. How does the product rule help in combining multiple logarithmic terms?
3. What other examples could we apply the quotient rule to?
4. What happens if the bases of the logarithms are not the same?
5. How do we handle expressions with radicals and logarithms?

### Tip
Always remember to check if you can apply logarithmic rules (product, quotient, power) to simplify expressions before performing calculations.

Condense the expression to a single logarithm. Write fractional exponents as radicals. Assume that all variables represent positive numbers. 3(log_3(3) + log_3(z)) - log_3(2) = ?

Learn how to condense the expression 3(log_3(3) + log_3(z)) - log_3(2) into a single logarithm. This problem involves logarithmic rules such as the product, power, and quotient rules, and is suitable for high school students in Grades 10-12.

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