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$4$
Calculate the value
$\left ( \log _{ 2 } { \left( 3 \right) } + \log _{ 4 } { \left( \color{#FF6800}{ 9 } \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$ $ Write the number in exponential form with base $ 3$
$\left ( \log _{ 2 } { \left( 3 \right) } + \log _{ 4 } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$\left ( \log _{ 2 } { \left( 3 \right) } + \log _{ \color{#FF6800}{ 4 } } { \left( 3 ^ { 2 } \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$ $ Write the number in exponential form with base $ 2$
$\left ( \log _{ 2 } { \left( 3 \right) } + \log _{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } } { \left( 3 ^ { 2 } \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$\left ( \log _{ 2 } { \left( 3 \right) } + \log _{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$ $ Simplify the expression using $ \log_{a^{y}}{b^{x}}=\dfrac{x}{y}\times\log_{a}{b}$
$\left ( \log _{ 2 } { \left( 3 \right) } + \color{#FF6800}{ \dfrac { \color{#FF6800}{ 2 } } { \color{#FF6800}{ 2 } } } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 3 } \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$\left ( \log _{ 2 } { \left( 3 \right) } + \color{#FF6800}{ \dfrac { \color{#FF6800}{ 2 } } { \color{#FF6800}{ 2 } } } \log _{ 2 } { \left( 3 \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$ $ Reduce the fraction $ $
$\left ( \log _{ 2 } { \left( 3 \right) } + \color{#FF6800}{ 1 } \log _{ 2 } { \left( 3 \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$\left ( \log _{ 2 } { \left( 3 \right) } + \color{#FF6800}{ 1 } \log _{ 2 } { \left( 3 \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$ $ Multiplying any number by 1 does not change the value $ $
$\left ( \log _{ 2 } { \left( 3 \right) } + \log _{ 2 } { \left( 3 \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$\left ( \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 3 } \right) } \color{#FF6800}{ + } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 3 } \right) } \right ) \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$ $ Calculate between similar terms $ $
$\color{#FF6800}{ 2 } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 3 } \right) } \left ( \log _{ 3 } { \left( 2 \right) } + \log _{ 9 } { \left( 4 \right) } \right )$
$\color{#FF6800}{ 2 } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 3 } \right) } \left ( \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 2 } \right) } \color{#FF6800}{ + } \log _{ \color{#FF6800}{ 9 } } { \left( \color{#FF6800}{ 4 } \right) } \right )$
$ $ Expand the expression to calculate the value $ $
$\color{#FF6800}{ 4 }$
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