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Formula
Calculate the value
$\log_{ \left(2 \sqrt{ 2 } \right) } {\left( \sqrt[ 4 ]{ 32 } \right)}$
$\dfrac { 5 } { 6 }$
Calculate the value
$\log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 32 } } \right) }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$\log _{ 2 \sqrt{ 2 } } { \left( \color{#FF6800}{ 2 } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \right) }$
$\log _{ \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { \left( \color{#FF6800}{ 2 } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \right) }$
 Simplify the expression using $\log_{a}{x\times y}=\log_{a}{x}+\log_{a}{y}$
$\log _{ \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { \left( \color{#FF6800}{ 2 } \right) } \color{#FF6800}{ + } \log _{ \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { \left( \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \right) }$
$\log _{ \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { \left( \color{#FF6800}{ 2 } \right) } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
 Rationalize the base of the logarithm 
$\color{#FF6800}{ 2 } \log _{ \color{#FF6800}{ 8 } } { \left( \color{#FF6800}{ 2 } \right) } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
$2 \log _{ 8 } { \left( \color{#FF6800}{ 2 } \right) } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
 Write the number in exponential form with base $2$
$2 \log _{ 2 ^ { 3 } } { \left( \color{#FF6800}{ 2 } \right) } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
$2 \log _{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 3 } } } { \left( \color{#FF6800}{ 2 } \right) } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
 Simplify the expression using $\log_{a^{y}}{a}=\dfrac{1}{y}\log_{a}{a}$
$2 \times \color{#FF6800}{ \dfrac { 1 } { 3 } } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
$2 \times \dfrac { 1 } { 3 } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
 The logarithm is equal to 1 if a base is same as an antilogarithm 
$2 \times \dfrac { 1 } { 3 } \times \color{#FF6800}{ 1 } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
$2 \times \dfrac { 1 } { 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
 Multiplying any number by 1 does not change the value 
$2 \times \dfrac { 1 } { 3 } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
$\color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 3 } } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
 Calculate the product of rational numbers 
$\color{#FF6800}{ \dfrac { 2 } { 3 } } + \log _{ 2 \sqrt{ 2 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
$\dfrac { 2 } { 3 } + \log _{ \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { \left( \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \right) }$
 Rationalize the base of the logarithm 
$\dfrac { 2 } { 3 } + \color{#FF6800}{ 2 } \log _{ \color{#FF6800}{ 8 } } { \left( \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \right) }$
$\dfrac { 2 } { 3 } + 2 \log _{ \color{#FF6800}{ 8 } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
 Write the number in exponential form with base $2$
$\dfrac { 2 } { 3 } + 2 \log _{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 3 } } } { \left( \sqrt[ 4 ]{ 2 } \right) }$
$\dfrac { 2 } { 3 } + 2 \log _{ \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 3 } } } { \left( \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \right) }$
 Simplify the expression using $\log_{a^{y}}{b}=\dfrac{1}{y}\times\log_{a}{b}$
$\dfrac { 2 } { 3 } + 2 \times \color{#FF6800}{ \dfrac { 1 } { 3 } } \log _{ \color{#FF6800}{ 2 } } { \left( \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \right) }$
$\dfrac { 2 } { 3 } + 2 \times \dfrac { 1 } { 3 } \log _{ 2 } { \left( \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \right) }$
 Convert the square root of the antilogarithm number of the logarithm to the power 
$\dfrac { 2 } { 3 } + 2 \times \dfrac { 1 } { 3 } \log _{ 2 } { \left( \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ \frac { 1 } { 4 } } } \right) }$
$\dfrac { 2 } { 3 } + 2 \times \dfrac { 1 } { 3 } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ \frac { 1 } { 4 } } } \right) }$
 Simplify the expression using $\log_{a}{a^{x}}=x\times\log_{a}{a}$
$\dfrac { 2 } { 3 } + 2 \times \dfrac { 1 } { 3 } \times \color{#FF6800}{ \dfrac { 1 } { 4 } } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) }$
$\dfrac { 2 } { 3 } + 2 \times \dfrac { 1 } { 3 } \times \dfrac { 1 } { 4 } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) }$
 The logarithm is equal to 1 if a base is same as an antilogarithm 
$\dfrac { 2 } { 3 } + 2 \times \dfrac { 1 } { 3 } \times \dfrac { 1 } { 4 } \times \color{#FF6800}{ 1 }$
$\dfrac { 2 } { 3 } + 2 \times \dfrac { 1 } { 3 } \times \dfrac { 1 } { 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 }$
 Multiplying any number by 1 does not change the value 
$\dfrac { 2 } { 3 } + 2 \times \dfrac { 1 } { 3 } \times \dfrac { 1 } { 4 }$
$\dfrac { 2 } { 3 } + \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 4 } }$
 Calculate the product of rational numbers 
$\dfrac { 2 } { 3 } + \color{#FF6800}{ \dfrac { 1 } { 6 } }$
$\color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } }$
 Find the sum of the fractions 
$\color{#FF6800}{ \dfrac { 5 } { 6 } }$
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