Given the function \( f(x) = \ln(e^x - 4) \), we want to find its inverse function, denoted as \( f^{-1}(x) \).

### Steps to Find the Inverse Function

1. **Start with the equation:**
   \[
   y = \ln(e^x - 4)
   \]
   
2. **Swap \( x \) and \( y \) to find the inverse:**
   \[
   x = \ln(e^y - 4)
   \]

3. **Exponentiate both sides to eliminate the natural logarithm:**
   \[
   e^x = e^y - 4
   \]

4. **Isolate \( e^y \):**
   \[
   e^y = e^x + 4
   \]

5. **Take the natural logarithm of both sides to solve for \( y \):**
   \[
   y = \ln(e^x + 4)
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \ln(e^x + 4)
\]

### Final Answer:
\[
f^{-1}(x) = \ln(e^x + 4)
\]

Would you like more details on how inverse functions work or have any other questions?

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Here are 5 related questions:
1. How do you verify if two functions are inverses of each other?
2. What are the domain and range of \( f(x) = \ln(e^x - 4) \)?
3. What is the derivative of the inverse function \( f^{-1}(x) \)?
4. How does the graph of a function compare to the graph of its inverse?
5. What is the significance of the natural logarithm and the exponential function in calculus?

**Tip:** Always check the domain of the original function before finding its inverse, as this will affect the domain of the inverse.

Learn how to find the inverse of the function f(x) = ln(e^x - 4) step-by-step. Explore the concepts of inverse functions, natural logarithms, and exponential functions with detailed explanations.

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