Find the invere of \[ y=e^{x+2} \]

To find the inverse of the function \( y = e^{x+2} \), we start by swapping \( x \) and \( y \) and then solve for \( y \): 1. Swap \( x \) and \( y \): \[ x = e^{y+2} \] 2. Take the natural logarithm of both sides: \[ \ln(x) = y + 2 \] 3. Solve for \( y \): \[ y = \ln(x) - 2 \] Thus, the inverse function is: \[ y = \ln(x) - 2 \] Feel free to explore the fascinating relationship between exponential and logarithmic functions! They essentially undo each other, making inverses a fun topic in algebraic manipulation. You might also find it interesting that the graph of a function and its inverse are reflections across the line \( y = x \). This reflection property can visually help in understanding how inverses operate, and you can even try plotting them to see this relationship!