The given equation is:

\[
y = 3 \log_4 (2x)
\]

This equation involves a logarithm with base 4. To understand it better, let's analyze and simplify it.

### Step-by-Step Simplification:

1. **Using the Change of Base Formula**:
   The change of base formula states:
   \[
   \log_b(x) = \frac{\log_a(x)}{\log_a(b)}
   \]
   where \( a \) is a new base (often base 10 or base \( e \)). So, we can rewrite the logarithmic expression with base 4 as:
   \[
   \log_4(2x) = \frac{\log(2x)}{\log(4)}
   \]

2. **Simplify the Logarithm**:
   Since \( \log(4) = 2\log(2) \), we have:
   \[
   \log_4(2x) = \frac{\log(2x)}{2\log(2)}
   \]

3. **Multiplying by 3**:
   Now, multiply the whole expression by 3:
   \[
   y = 3 \cdot \frac{\log(2x)}{2\log(2)} = \frac{3 \log(2x)}{2\log(2)}
   \]

This is a simplified form of the given equation. You can further analyze it based on the value of \( x \).

Would you like to explore more about this equation or any specific aspect?

### Related Questions:
1. How would you find the derivative of \( y = 3\log_4(2x) \)?
2. Can you solve for \( x \) when \( y = 0 \)?
3. How does the graph of \( y = 3\log_4(2x) \) look compared to a regular logarithmic graph?
4. What is the behavior of \( y \) as \( x \) approaches 0?
5. How does changing the base of the logarithm affect the equation?

**Tip**: Logarithmic functions grow very slowly compared to exponential functions, which is useful when analyzing graphs or comparing growth rates.

Understand how to simplify the logarithmic equation y = 3log4(2x) using the change of base formula. This detailed step-by-step solution explores key algebraic concepts, including logarithmic transformations. Suitable for students in Grades 10-12.

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