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$2$
Calculate the value
$\log _{ \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } } { \left( \dfrac { 1 } { 4 } \right) }$
$ $ 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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