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Calculate the value
Answer
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$- \dfrac { 1 } { 2 }$
Calculate the value
$\log _{ \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 5 } } } } { \left( \sqrt{ 5 } \right) }$
$ $ 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 $ $
$- \log _{ 5 } { \left( \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } } \right) }$
$- \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } } \right) }$
$ $ Simplify the expression using $ \log_{a}{a^{x}}=x\times\log_{a}{a}$
$- \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } \right) } \right )$
$- \left ( \dfrac { 1 } { 2 } \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } \right) } \right )$
$ $ 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 $ $
$- \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } }$
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