Answer
$x=1$
Work Step by Step
With the bases on both sides of the equation the same, the exponents of the given equation, $
e^{\ln2x}=e^{\ln(x+1)}
$ can be equated. That is,
\begin{align*}\require{cancel}
\ln2x&=\ln(x+1)
.\end{align*}
Since $\ln a=\ln b$ implies $a=b$, the equation above is equivalent to
\begin{align*}\require{cancel}
2x&=x+1
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
2x-x&=1
\\
x&=1
.\end{align*}
Hence, the solution to the equation $
e^{\ln2x}=e^{\ln(x+1)}
$ is $
x=1
$.
