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.