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$2$
Calculate the value
$\log _{ 5 } { \left( 7 \right) } \log _{ \color{#FF6800}{ 7 } } { \left( \color{#FF6800}{ 25 } \right) }$
$ $ Use $ \log_{a}{x}=\dfrac{\log_{b}{x}}{\log_{b}{a}} $ to change a number of the logarithmic base $ $
$\log _{ 5 } { \left( 7 \right) } \times \color{#FF6800}{ \dfrac { \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 25 } \right) } } { \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 7 } \right) } } }$
$\log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 7 } \right) } \times \dfrac { \log _{ 5 } { \left( 25 \right) } } { \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 7 } \right) } }$
$ $ Reduce the expression with $ \log _{ 5 } { \left( 7 \right) } $ , the greatest common devisor $ $
$\log _{ 5 } { \left( 25 \right) }$
$\log _{ 5 } { \left( \color{#FF6800}{ 25 } \right) }$
$ $ Write the number in exponential form with base $ 5$
$\log _{ 5 } { \left( \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 2 } } \right) }$
$\log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 2 } } \right) }$
$ $ Simplify the expression using $ \log_{a}{a^{x}}=x\times\log_{a}{a}$
$\color{#FF6800}{ 2 } \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } \right) }$
$2 \log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ 5 } \right) }$
$ $ 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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