Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents. \[ e^{x+2}=\frac{1}{e} \]

Step 1 ：Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents: \(e^{x+2}=\frac{1}{e}\).

Step 2 ：The equation can be rewritten as \(e^{x+2} = e^{-1}\). This implies that \(x+2 = -1\).

Step 3 ：Solving for x will give us the solution: \(x = -1 - 2\).

Step 4 ：The solution to the equation is \(x = -3\). This is because when we equate the exponents, we get \(x + 2 = -1\). Solving for x gives us \(x = -3\).

Step 5 ：Final Answer: \(\boxed{-3}\)