To simplify the expression \( \log_e^2((e^{4a}) + a e^{4a}) \), let's proceed step-by-step.

The expression can be rewritten as:
\[
\log_e^2((e^{4a}) + a e^{4a})
\]

This represents the square of the natural logarithm (denoted as \( \ln \)) of the expression \( (e^{4a} + a e^{4a}) \).

1. First, let's factor out \( e^{4a} \) from the expression inside the logarithm:
   \[
   e^{4a} + a e^{4a} = e^{4a}(1 + a)
   \]

2. Now, substitute this back into the logarithm:
   \[
   \log_e^2(e^{4a}(1 + a)) = \left( \ln(e^{4a}(1 + a)) \right)^2
   \]

3. Using the logarithm property \( \ln(xy) = \ln(x) + \ln(y) \), we can split the logarithm:
   \[
   \ln(e^{4a}(1 + a)) = \ln(e^{4a}) + \ln(1 + a)
   \]

4. The logarithm of \( e^{4a} \) simplifies to:
   \[
   \ln(e^{4a}) = 4a
   \]

   So the expression becomes:
   \[
   \ln(e^{4a}(1 + a)) = 4a + \ln(1 + a)
   \]

5. Now, square this result:
   \[
   \left( 4a + \ln(1 + a) \right)^2
   \]

This is the simplified form of the original expression.

### Final Answer:
\[
\left( 4a + \ln(1 + a) \right)^2
\]

Would you like to explore the steps further or ask any questions?

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Here are five related questions:

1. What are the logarithmic properties for simplifying products and quotients?
2. How does the natural logarithm behave with exponential functions like \( e^{x} \)?
3. What happens when \( a = 0 \) in the expression \( \ln(e^{4a}(1 + a)) \)?
4. How can the binomial expansion be used to approximate \( \ln(1 + a) \)?
5. How do you differentiate or integrate logarithmic functions like \( \ln(1 + a) \)?

**Tip:** Always remember the key logarithmic identities when simplifying expressions, such as \( \ln(ab) = \ln(a) + \ln(b) \) and \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).

Discover how to simplify the expression log_e^2((e^4a) + ae^4a) using logarithmic properties and exponential functions. This step-by-step solution breaks down the process, applying key logarithmic formulas. Ideal for students in Grades 10-12.

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