Question - Finding the Inverse of an Exponential Function

Let \( y = e^{4x} - 5 \)

Swap x and y to find the inverse: \( x = e^{4y} - 5 \)

Add 5 to both sides: \( x + 5 = e^{4y} \)

Take the natural logarithm of both sides: \( \ln(x + 5) = \ln(e^{4y}) \)

Use the property of logarithms: \( \ln(x + 5) = 4y \)

Divide by 4: \( y = \frac{1}{4}\ln(x + 5) \)

The inverse function is \( f^{-1}(x) = \frac{1}{4}\ln(x + 5) \)