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Calculate the value
$\log_{ \left( \frac{ 1 }{ 5 } \right) } {\left( \sqrt{ 5 } \right)}$
$- \dfrac { 1 } { 2 }$
Calculate the value
$\log _{ \color{#FF6800}{ \frac { 1 } { 5 } } } { \left( \sqrt{ 5 } \right) }$
 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 
$\color{#FF6800}{ - } \log _{ \color{#FF6800}{ 5 } } { \left( \sqrt{ 5 } \right) }$
$- \log _{ 5 } { \left( \sqrt{ \color{#FF6800}{ 5 } } \right) }$
 Convert the square root of the antilogarithm number of the logarithm to the power 
$- \log _{ 5 } { \left( \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ \frac { 1 } { 2 } } } \right) }$
$- \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ \frac { 1 } { 2 } } } \right) }$
 Simplify the expression using $\log_{a}{a^{x}}=x\times\log_{a}{a}$
$- \left ( \color{#FF6800}{ \dfrac { 1 } { 2 } } \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } \right) } \right )$
$- \left ( \dfrac { 1 } { 2 } \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } \right) } \right )$
 The logarithm is equal to 1 if a base is same as an antilogarithm 
$- \left ( \dfrac { 1 } { 2 } \times \color{#FF6800}{ 1 } \right )$
$- \left ( \dfrac { 1 } { 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \right )$
 Multiplying any number by 1 does not change the value 
$- \color{#FF6800}{ \dfrac { 1 } { 2 } }$
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