# Solve for: e^{3x}-7e^x+6=0

## Expression: ${e}^{3x}-7{e}^{x}+6=0$

Use ${a}^{mn}={\left( {a}^{n} \right)}^{m}$ to transform the expression

${\left( {e}^{x} \right)}^{3}-7{e}^{x}+6=0$

To get an equation that is easier to solve, substitute $t$ for ${e}^{x}$

${t}^{3}-7t+6=0$

Solve the equation for $t$

$\begin{array} { l }t=1,\\t=-3,\\t=2\end{array}$

Substitute back $t={e}^{x}$

$\begin{array} { l }{e}^{x}=1,\\{e}^{x}=-3,\\{e}^{x}=2\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=0,\\{e}^{x}=-3,\\{e}^{x}=2\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=0,\\x\notin ℝ,\\{e}^{x}=2\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=0,\\x\notin ℝ,\\x=\ln\left({2}\right)\end{array}$

Find the union

$\begin{array} { l }x=0,\\x=\ln\left({2}\right)\end{array}$

The equation has $2$ solutions

\begin{align*}&\begin{array} { l }x_1=0,& x_2=\ln\left({2}\right)\end{array} \\&\begin{array} { l }x_1=0,& x_2\approx0.693147\end{array}\end{align*}

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