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$\log_{ 8 } {\left( 2 \sqrt{ 2 } \right)}$
$\dfrac { 1 } { 2 }$
Calculate the value
$\log _{ \color{#FF6800}{ 8 } } { \left( 2 \sqrt{ 2 } \right) }$
$ $ Write the number in exponential form with base $ 2$
$\log _{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 3 } } } { \left( 2 \sqrt{ 2 } \right) }$
$\log _{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 3 } } } { \left( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right) }$
$ $ Simplify the expression using $ \log_{a^{y}}{b}=\dfrac{1}{y}\times\log_{a}{b}$
$\color{#FF6800}{ \dfrac { 1 } { 3 } } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right) }$
$\dfrac { 1 } { 3 } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \right) }$
$ $ Simplify the expression using $ \log_{a}{x\times y}=\log_{a}{x}+\log_{a}{y}$
$\dfrac { 1 } { 3 } \left ( \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) } \color{#FF6800}{ + } \log _{ \color{#FF6800}{ 2 } } { \left( \sqrt{ \color{#FF6800}{ 2 } } \right) } \right )$
$\dfrac { 1 } { 3 } \left ( \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) } + \log _{ 2 } { \left( \sqrt{ 2 } \right) } \right )$
$ $ The logarithm is equal to 1 if a base is same as an antilogarithm $ $
$\dfrac { 1 } { 3 } \left ( \color{#FF6800}{ 1 } + \log _{ 2 } { \left( \sqrt{ 2 } \right) } \right )$
$\dfrac { 1 } { 3 } \left ( 1 + \log _{ 2 } { \left( \sqrt{ \color{#FF6800}{ 2 } } \right) } \right )$
$ $ Convert the square root of the antilogarithm number of the logarithm to the power $ $
$\dfrac { 1 } { 3 } \left ( 1 + \log _{ 2 } { \left( \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ \frac { 1 } { 2 } } } \right) } \right )$
$\dfrac { 1 } { 3 } \left ( 1 + \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ \frac { 1 } { 2 } } } \right) } \right )$
$ $ Simplify the expression using $ \log_{a}{a^{x}}=x\times\log_{a}{a}$
$\dfrac { 1 } { 3 } \left ( 1 + \color{#FF6800}{ \dfrac { 1 } { 2 } } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) } \right )$
$\dfrac { 1 } { 3 } \left ( 1 + \dfrac { 1 } { 2 } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) } \right )$
$ $ The logarithm is equal to 1 if a base is same as an antilogarithm $ $
$\dfrac { 1 } { 3 } \left ( 1 + \dfrac { 1 } { 2 } \times \color{#FF6800}{ 1 } \right )$
$\dfrac { 1 } { 3 } \left ( 1 + \dfrac { 1 } { 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \right )$
$ $ Multiplying any number by 1 does not change the value $ $
$\dfrac { 1 } { 3 } \left ( 1 + \color{#FF6800}{ \dfrac { 1 } { 2 } } \right )$
$\dfrac { 1 } { 3 } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 2 } } \right )$
$ $ Add two numbers $ 1 $ and $ \dfrac { 1 } { 2 }$
$\dfrac { 1 } { 3 } \times \color{#FF6800}{ \dfrac { 3 } { 2 } }$
$\color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 3 } { 2 } }$
$ $ Calculate the product of rational numbers $ $
$\color{#FF6800}{ \dfrac { 1 } { 2 } }$
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