# Evaluate: 9^x-7 * 3^x+10=0

## Expression: ${9}^{x}-7 \times {3}^{x}+10=0$

Write the number in exponential form with the base of $3$

${\left( {3}^{2} \right)}^{x}-7 \times {3}^{x}+10=0$

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

${\left( {3}^{x} \right)}^{2}-7 \times {3}^{x}+10=0$

To get an equation that is easier to solve, substitute $t$ for ${3}^{x}$

${t}^{2}-7t+10=0$

Solve the equation for $t$

$\begin{array} { l }t=2,\\t=5\end{array}$

Substitute back $t={3}^{x}$

$\begin{array} { l }{3}^{x}=2,\\{3}^{x}=5\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=\log_{ 3 }({ 2 }),\\{3}^{x}=5\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=\log_{ 3 }({ 2 }),\\x=\log_{ 3 }({ 5 })\end{array}$

The equation has $2$ solutions

\begin{align*}&\begin{array} { l }x_1=\log_{ 3 }({ 2 }),& x_2=\log_{ 3 }({ 5 })\end{array} \\&\begin{array} { l }x_1\approx0.63093,& x_2\approx1.46497\end{array}\end{align*}

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