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

Explanation:

Step1: Rewrite the right side

Recall that $\frac{1}{e}=e^{-1}$ since $a^{-n}=\frac{1}{a^{n}}$. So the equation $e^{x + 8}=\frac{1}{e}$ becomes $e^{x+8}=e^{-1}$.

Step2: Equate the exponents

For exponential functions with the same base $a$ (here $a = e>0,a
eq1$), if $a^{m}=a^{n}$, then $m = n$. So we set $x + 8=-1$.

Step3: Solve for x

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

Answer:

$\{-9\}$