Question

solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
e^{x + 8} = \frac{1}{e}
the solution set is \boxed{}

Explanation:

Step1: Rewrite the right side

We know that \(\frac{1}{e}=e^{-1}\), so the equation \(e^{x + 8}=\frac{1}{e}\) can be rewritten as \(e^{x + 8}=e^{-1}\).

Step2: Equate the exponents

Since the bases are the same (\(e\)) and the exponential function \(y = e^{x}\) is one - to - one (if \(e^{a}=e^{b}\), then \(a = b\)), we can set the exponents equal to each other: \(x+8=-1\).

Step3: Solve for \(x\)

Subtract 8 from both sides of the equation \(x + 8=-1\). We get \(x=-1 - 8=-9\).

Answer:

The solution set is \(\{-9\}\)