Let's solve the given problems using the definition of the natural logarithmic function. Here's how:

Part (a):

The equation is:
$\ln(x) = 4$

The natural logarithm $\ln(x)$ is the logarithm to the base $e$. To solve for $x$, rewrite the equation in exponential form:
$x = e^4$

Thus, the solution is:
$x = e^4$

Part (b):

The equation is:
$\ln(e^5) = x$

Using the property of logarithms, $\ln(e^a) = a$, this simplifies directly:
$x = 5$

Thus, the solution is:
$x = 5$

Final Answers:

(a) $x = e^4$
(b) $x = 5$

Would you like a further explanation of the logarithmic properties used?

Here are 5 related questions to deepen your understanding:

1. How do you convert between logarithmic and exponential forms?
2. What is the value of $e$, and why is it significant in logarithms?
3. How do the properties of natural logarithms (e.g., $\ln(ab) = \ln(a) + \ln(b)$) simplify solving equations?
4. Can $\ln(x)$ ever be negative? Why or why not?
5. How does $\ln(x)$ relate to other logarithmic functions, like $\log_{10}(x)$?

Tip: Always remember that $\ln(x)$ is undefined for $x \leq 0$, as the logarithm is only defined for positive real numbers!