Let's work through the problem:

Part (a): Rewrite $\ln 8 = y$ as an exponential equation.

The natural logarithm equation $\ln 8 = y$ means: $\ln 8 = \log_e 8$ Using the definition of logarithms, the exponential form is: $e^y = 8$

Part (b): Rewrite $e^x = 3$ as a logarithmic equation.

The exponential equation $e^x = 3$ can be rewritten using the natural logarithm definition: $x = \ln 3$

Final Answers:

(a) $e^y = 8$
(b) $x = \ln 3$

Let me know if you'd like further clarifications!

Related Questions:

1. How is the natural logarithm ($\ln$) related to base-$e$ exponentials?
2. What are the properties of logarithms that can simplify expressions?
3. Can $\ln$ equations always be converted into exponential form?
4. How do you solve equations involving $\ln$ and $e^x$ together?
5. What are some real-life applications of natural logarithms?

Tip:

To verify transformations between logarithmic and exponential forms, always check the base and ensure the equation remains balanced.