Answer
$x\approx-10.718$
Work Step by Step
Taking the natural logarithm of both sides, the given equation, $
e^{-0.205x}=9
$ is equivalent to
\begin{align*}\require{cancel}
\ln e^{-0.205x}&=\ln9
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
-0.205x(\ln e)&=\ln9
&(\text{use }\log_b x^y=y\log_b x)
\\
-0.205x(1)&=\ln9
&(\text{use }\ln e=1)
\\
-0.205x&=\ln9
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\dfrac{\cancel{-0.205}x}{\cancel{-0.205}}&=\dfrac{\ln9}{-0.205}
\\\\
x&=\dfrac{\ln9}{-0.205}
\\\\
x&\approx-10.718
.\end{align*}
Hence, the solution to the equation $
e^{-0.205x}=9
$ is $
x\approx-10.718
$.
