To find the inverse of the function \( f(x) = \ln(e^x - 5) \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \):
\[
y = \ln(e^x - 5)
\]

### Step 2: Swap \( x \) and \( y \) to find the inverse:
\[
x = \ln(e^y - 5)
\]

### Step 3: Exponentiate both sides to eliminate the natural logarithm:
\[
e^x = e^y - 5
\]

### Step 4: Solve for \( e^y \):
\[
e^y = e^x + 5
\]

### Step 5: Take the natural logarithm of both sides:
\[
y = \ln(e^x + 5)
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \ln(e^x + 5)
\]

Would you like further details on the steps? Here are five related questions for you:

1. What are the domain and range of the original function?
2. How would you verify the inverse function?
3. Can you find the derivative of the inverse function?
4. What happens if we change the function to \( f(x) = \ln(e^x - c) \)?
5. How do inverse functions behave under composition?

**Tip:** Inverse functions swap the roles of inputs and outputs, meaning \( f(f^{-1}(x)) = x \).

Learn how to find the inverse of the logarithmic function f(x) = ln(e^x - 5) with a clear, step-by-step explanation. This solution covers key concepts like logarithmic and exponential functions, their inverse relationships, and the necessary algebraic manipulations. Ideal for high school students in Grades 10-12.

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