To solve the expression involving the natural logarithm and exponentiation, $\ln(e^3)$, let's break it down:

Step-by-Step Explanation:

1. Recall the relationship between logarithms and exponents: The natural logarithm $\ln(x)$ is the inverse of the exponential function $e^x$. Specifically, $\ln(e^x) = x$.

2. Apply this property: In the given expression $\ln(e^3)$, the argument of the logarithm is $e^3$. By the property mentioned, we can simplify $\ln(e^3)$ to:

   $\ln(e^3) = 3$

Final Answer:

$\ln(e^3) = 3$

Let me know if you need more details or further explanation!

Here are five related questions:

1. What is $\ln(e^5)$?
2. How would you simplify $\ln(e^{x+2})$?
3. What is the value of $\ln(1)$?
4. Can you explain how $e^{\ln(x)}$ simplifies for any $x$?
5. How does the base of the logarithm affect the simplification of $\ln(e^x)$?

Tip: Remember, the natural logarithm $\ln(x)$ is shorthand for logarithm with base $e$, where $e \approx 2.718$.