Answer
$x=\dfrac{2}{3}$
Work Step by Step
With the bases on both sides of the equation the same, the exponents of the given equation, $
e^{\ln(6-x)}=e^{\ln(4+2x)}
$ can be equated. That is,
\begin{align*}\require{cancel}
\ln(6-x)&=\ln(4+2x)
.\end{align*}
Since $\ln a=\ln b$ implies $a=b$, the equation above is equivalent to
\begin{align*}\require{cancel}
6-x&=4+2x
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
6-4&=2x+x
\\
2&=3x
\\\\
\dfrac{2}{3}&=\dfrac{\cancel3x}{\cancel3}
\\\\
\dfrac{2}{3}&=x
\\\\
x&=\dfrac{2}{3}
.\end{align*}
Hence, the solution to the equation $
e^{\ln(6-x)}=e^{\ln(4+2x)}
$ is $
x=\dfrac{2}{3}
$.
