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.