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The given expression is:

\[
\ln(e^4) \cdot \lg(100)
\]

Let me solve this step-by-step.

### Step 1: Simplifying \(\ln(e^4)\)
The natural logarithm of \(e^x\) simplifies to \(x\) because \(\ln(e^x) = x\).

\[
\ln(e^4) = 4
\]

### Step 2: Simplifying \(\lg(100)\)
\(\lg(x)\) refers to the logarithm base 10 of \(x\). Since \(100 = 10^2\), the logarithm of 100 is:

\[
\lg(100) = 2
\]

### Step 3: Multiplying the results
Now, multiply the simplified values:

\[
4 \cdot 2 = 8
\]

### Final Answer:

\[
\ln(e^4) \cdot \lg(100) = 8
\]

Let me know if you'd like more details on any of these steps!

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Here are 5 related questions to explore:
1. What is the difference between natural logarithms and common logarithms?
2. How would the expression change if it were \(\ln(e^x) \cdot \lg(10^3)\)?
3. How can logarithms be used to simplify complex exponential expressions?
4. What are the properties of logarithms that help in solving equations like this?
5. How can we calculate logarithms for non-integer values?

**Tip**: When dealing with logarithms, remember that \(\ln(e^x) = x\) and \(\lg(10^x) = x\), which makes them very useful for simplifying exponential expressions!

Learn how to simplify the logarithmic expression ln(e^4) * lg(100) using properties of logarithms and exponents. The problem covers natural and common logarithms and is suitable for students in Grades 10-12.

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