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$4$
Calculate the value
$\log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 2 } } } \right) } + \color{#FF6800}{ 2 } \log _{ \color{#FF6800}{ 3 } } { \left( \sqrt{ \color{#FF6800}{ 54 } } \right) }$
$ $ Calculate addition of logarithm $ $
$\log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 81 } \right) }$
$\log _{ 3 } { \left( \color{#FF6800}{ 81 } \right) }$
$ $ Write the number in exponential form with base $ 3$
$\log _{ 3 } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 4 } } \right) }$
$\log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 4 } } \right) }$
$ $ Simplify the expression using $ \log_{a}{a^{x}}=x\times\log_{a}{a}$
$\color{#FF6800}{ 4 } \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } \right) }$
$4 \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } \right) }$
$ $ The logarithm is equal to 1 if a base is same as an antilogarithm $ $
$4 \times \color{#FF6800}{ 1 }$
$4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 }$
$ $ Multiplying any number by 1 does not change the value $ $
$\color{#FF6800}{ 4 }$
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