Answer
$x\approx0.563$
Work Step by Step
Using the properties of equality, the given equation, $
19=2e^{4x}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{19}{2}=e^{4x}
.\end{array}
Taking the natural logarithm of both sides and using the properties of logarithms, the value of the variable that satisfies the equation, $
\dfrac{19}{2}=e^{4x}
,$ is
\begin{array}{l}\require{cancel}
\ln\dfrac{19}{2}=\ln e^{4x}
\\\\
\ln\dfrac{19}{2}=4x(\ln e)
\\\\
\ln\dfrac{19}{2}=4x(1)
\\\\
\ln\dfrac{19}{2}=4x
\\\\
\dfrac{\ln\dfrac{19}{2}}{4}=x
\\\\
x\approx0.563
.\end{array}
