We can simplify the expression \( 3 \log 54 + 3 \log 18 - 3 \log 12 \) using properties of logarithms.

### Step-by-step process:

1. **Factor out the common coefficient:**

   Since all the terms have a factor of 3, we can factor out 3:
   \[
   3 (\log 54 + \log 18 - \log 12)
   \]

2. **Apply the logarithmic properties:**
   
   - Sum of logarithms: \( \log a + \log b = \log(ab) \)
   - Difference of logarithms: \( \log a - \log b = \log\left(\frac{a}{b}\right) \)

   Using these properties, simplify the expression inside the parentheses:
   \[
   \log 54 + \log 18 - \log 12 = \log \left( \frac{54 \times 18}{12} \right)
   \]

3. **Simplify the expression inside the logarithm:**

   First, calculate the multiplication and division inside the logarithm:
   \[
   \frac{54 \times 18}{12} = \frac{972}{12} = 81
   \]

   So the expression becomes:
   \[
   3 \log 81
   \]

4. **Simplify further:**

   We know that \( \log 81 = \log (3^4) = 4 \log 3 \), so:
   \[
   3 \log 81 = 3 \times 4 \log 3 = 12 \log 3
   \]

Thus, the simplified expression is:
\[
12 \log 3
\]

Would you like further details or have any questions?

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Here are some related questions:

1. How do the properties of logarithms simplify complex expressions?
2. Can logarithmic properties be applied to logarithms with different bases?
3. What are common mistakes when applying logarithmic properties?
4. How can we use logarithms in solving real-world problems, like exponential growth?
5. What is the relationship between logarithms and exponents?

**Tip:** Always look for common factors or properties that can simplify logarithmic expressions.

Learn how to simplify the logarithmic expression 3Log 54 + 3Log 18 - 3Log 12 using properties of logarithms. This step-by-step solution explains the use of logarithmic rules, including the sum and difference of logarithms. Suitable for students in Grades 10-12.

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