https://file.aiquickdraw.com/mathbot/file/17288534911825t2gs9ug.jpg

Let's solve problem 30 from the image step by step.

### Problem: 
\[
\log_3\left(\frac{1}{9}\right) - 3\log_3(3)
\]

### Step 1: Apply logarithmic properties.

1. **Simplify** \(\log_3\left(\frac{1}{9}\right)\).

   We know that \(\frac{1}{9} = 3^{-2}\). Using the logarithmic property \(\log_b(a^n) = n \log_b(a)\), we get:
   \[
   \log_3\left(\frac{1}{9}\right) = \log_3(3^{-2}) = -2 \log_3(3)
   \]
   Since \(\log_3(3) = 1\), this simplifies to:
   \[
   -2 \log_3(3) = -2 \cdot 1 = -2
   \]

2. **Simplify** \(3\log_3(3)\).

   From the logarithmic property \(n \log_b(a) = \log_b(a^n)\), and knowing that \(\log_3(3) = 1\), we get:
   \[
   3 \log_3(3) = 3 \cdot 1 = 3
   \]

### Step 2: Combine the results.

Now, we can subtract the two terms:
\[
-2 - 3 = -5
\]

### Final Answer:
\[
\log_3\left(\frac{1}{9}\right) - 3\log_3(3) = -5
\]

Would you like more details on any step, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. How would you evaluate \(\log_3\left(\frac{27}{9}\right)\) using properties of logarithms?
2. Can you simplify \(\log_b\left(\frac{1}{b^n}\right)\) in general terms?
3. What happens when you evaluate \(\log_b(1)\)? Why is it always 0?
4. How can you use the change-of-base formula to evaluate logarithms with bases other than 10 or \(e\)?
5. How would you simplify \(n \log_b(b)\) for any \(n\)?

### Tip:
Whenever you see a logarithm of a fraction, try expressing the fraction in terms of powers of the base, as this often simplifies the problem quickly.

Learn how to evaluate log_3(1/9) - 3log_3(3) step-by-step by applying logarithmic properties. Understand the simplification of logarithms involving powers of a base and how to handle logarithmic subtraction. Suitable for high school students in Grades 10-12.

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