Question

solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
$e^{x + 7} = \frac{1}{e}$
the solution set is \boxed{}

Explanation:

Step1: Rewrite the right side

Recall that $\frac{1}{e}=e^{-1}$, so the equation $e^{x + 7}=\frac{1}{e}$ becomes $e^{x + 7}=e^{-1}$.

Step2: Equate the exponents

Since the bases are the same ($e$) and the exponential function is one - to - one, if $e^{a}=e^{b}$, then $a = b$. So we set $x + 7=-1$.

Step3: Solve for x

Subtract 7 from both sides of the equation $x+7=-1$. We get $x=-1 - 7=-8$.

Answer:

$\{-8\}$