Answer
$x\approx-0.485$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
e^{2-x}=12
,$ take the natural logarithm of both sides. Then use the laws of logarithms and the properties of equality to isolate the variable. Express the answer with $3$ decimal places.
$\bf{\text{Solution Details:}}$
Taking the natural logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\ln e^{2-x}=\ln 12
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent
\begin{array}{l}\require{cancel}
(2-x)\ln e=\ln 12
.\end{array}
Since $\ln e =1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(2-x)(1)=\ln 12
\\\\
2-x=\ln 12
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
-x=-2+\ln 12
\\\\
-1(-x)=-1(-2+\ln 12)
\\\\
x=2-\ln 12
\\\\
x\approx-0.485
.\end{array}
