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Formula
Calculate the value
$\log_{ \left( \frac{ 1 }{ 2 } \right) } {\left( \dfrac{ 1 }{ 4 } \right)}$
$2$
Calculate the value
$\log _{ \color{#FF6800}{ \frac { 1 } { 2 } } } { \left( \dfrac { 1 } { 4 } \right) }$
 Since the bottom of a logarithm is a fraction with a numerator of 1, put a minus to the logarithm and take an inverse number for the bottom 
$\color{#FF6800}{ - } \log _{ \color{#FF6800}{ 2 } } { \left( \dfrac { 1 } { 4 } \right) }$
$- \log _{ 2 } { \left( \dfrac { \color{#FF6800}{ 1 } } { 4 } \right) }$
 Since the logarithm of antilogarithm numbers and numerator is 1 as the fraction, add minus to the logarithm and take reciprocal to antilogarithm numbers 
$- \left ( \color{#FF6800}{ - } \log _{ 2 } { \left( \color{#FF6800}{ 4 } \right) } \right )$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \log _{ 2 } { \left( 4 \right) } \right )$
 Simplify Minus 
$\log _{ 2 } { \left( 4 \right) }$
$\log _{ 2 } { \left( \color{#FF6800}{ 4 } \right) }$
 Write the number in exponential form with base $2$
$\log _{ 2 } { \left( \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \right) }$
$\log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \right) }$
 Simplify the expression using $\log_{a}{a^{x}}=x\times\log_{a}{a}$
$\color{#FF6800}{ 2 } \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) }$
$2 \log _{ \color{#FF6800}{ 2 } } { \left( \color{#FF6800}{ 2 } \right) }$
 The logarithm is equal to 1 if a base is same as an antilogarithm 
$2 \times \color{#FF6800}{ 1 }$
$2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 }$
 Multiplying any number by 1 does not change the value 
$\color{#FF6800}{ 2 }$
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