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Answer
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$- \dfrac { 1 } { 4 }$
Calculate the value
$\log _{ 10 } { \left( \dfrac { \color{#FF6800}{ 1 } } { \sqrt[ 4 ]{ 10 } } \right) }$
$ $ Since the logarithm of antilogarithm numbers and numerator is 1 as the fraction, add minus to the logarithm and take reciprocal to antilogarithm numbers $ $
$\color{#FF6800}{ - } \log _{ 10 } { \left( \sqrt[ 4 ]{ 10 } \right) }$
$- \log _{ 10 } { \left( \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 10 } } \right) }$
$ $ Convert the square root of the antilogarithm number of the logarithm to the power $ $
$- \log _{ 10 } { \left( \color{#FF6800}{ 10 } ^ { \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } } \right) }$
$- \log _{ \color{#FF6800}{ 10 } } { \left( \color{#FF6800}{ 10 } ^ { \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } } \right) }$
$ $ Simplify the expression using $ \log_{a}{a^{x}}=x\times\log_{a}{a}$
$- \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \log _{ \color{#FF6800}{ 10 } } { \left( \color{#FF6800}{ 10 } \right) } \right )$
$- \left ( \dfrac { 1 } { 4 } \log _{ \color{#FF6800}{ 10 } } { \left( \color{#FF6800}{ 10 } \right) } \right )$
$ $ The logarithm is equal to 1 if a base is same as an antilogarithm $ $
$- \left ( \dfrac { 1 } { 4 } \times \color{#FF6800}{ 1 } \right )$
$- \left ( \dfrac { 1 } { 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \right )$
$ $ Multiplying any number by 1 does not change the value $ $
$- \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } }$
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