We have the expression:

\[
\log_{e^2}\left( e^4 \cdot a + a \cdot e^4 \cdot a \right)
\]

### Step 1: Simplify the terms inside the logarithm

First, factor out common terms from the expression inside the logarithm. We notice that both terms \( e^4 \cdot a \) and \( a \cdot e^4 \cdot a \) contain \( e^4 \cdot a \) as a common factor:

\[
e^4 \cdot a + a \cdot e^4 \cdot a = e^4 \cdot a (1 + a)
\]

So, the expression becomes:

\[
\log_{e^2}(e^4 \cdot a (1 + a))
\]

### Step 2: Break down the logarithm using logarithmic properties

Now, using the property \( \log_b(x \cdot y) = \log_b(x) + \log_b(y) \), we can break the logarithm down:

\[
\log_{e^2}(e^4 \cdot a \cdot (1 + a)) = \log_{e^2}(e^4) + \log_{e^2}(a) + \log_{e^2}(1 + a)
\]

### Step 3: Simplify the logarithms

We know that \( \log_b(b^x) = x \), so:

\[
\log_{e^2}(e^4) = \frac{4}{2} = 2
\]

Thus, the expression becomes:

\[
2 + \log_{e^2}(a) + \log_{e^2}(1 + a)
\]

This is the simplified expression for \( \log_{e^2}(e^4 \cdot a + a \cdot e^4 \cdot a) \).

### Final Answer:
\[
2 + \log_{e^2}(a) + \log_{e^2}(1 + a)
\]

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Would you like any further details on this process? Here are a few related questions:

1. How do logarithmic properties work in base transformations?
2. How can logarithms simplify complex expressions in algebra?
3. What are some applications of logarithms in real-world problems?
4. Can you explain how to handle logarithms with different bases?
5. What is the difference between natural logs and logs of other bases?

**Tip**: When simplifying logarithmic expressions, always look for common factors or use properties like the product, quotient, or power rules!

This detailed explanation simplifies the expression log_e^2(e^4*a + a*e^4*a) using logarithmic properties such as the product and exponent rules. Learn how to break down complex logarithmic expressions and factor common terms. Suitable for students in Grades 10-12.

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