To find the inverse of the function $f(x) = e^{7x} - 5$, follow these steps:

Step 1: Replace $f(x)$ with $y$

$y = e^{7x} - 5$

Step 2: Solve for $x$

1. Add 5 to both sides: $y + 5 = e^{7x}$

2. Take the natural logarithm (ln) of both sides to remove the exponential: $\ln(y + 5) = 7x$

3. Solve for $x$ by dividing both sides by 7: $x = \frac{\ln(y + 5)}{7}$

Step 3: Replace $y$ with $x$ to express the inverse function

$f^{-1}(x) = \frac{\ln(x + 5)}{7}$

Thus, the inverse function is: $f^{-1}(x) = \frac{\ln(x + 5)}{7}$

Let me know if you'd like more details on any step or clarification!

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Here are five related questions:

1. How do you verify that a function and its inverse are correct?
2. What is the domain of the inverse function $f^{-1}(x)$?
3. How does the natural logarithm function work with exponential functions?
4. What happens if you try to find the inverse of $f(x) = e^{kx} - c$, where $k$ and $c$ are constants?
5. How does this process change if the base of the exponential function is not $e$?

Tip: When finding inverses of functions, always check that the original function is one-to-one (i.e., passes the horizontal line test). This ensures that the function has an inverse.