Simple Logarithm

Take note: $\Large b^{\log_{b}x}=x$

$\LARGE 4^{ \log_{16}25}$ $\LARGE =4^{ \log_{16}5^2}$ $\LARGE= 4^{2 \log_{16}5}$ $\LARGE= 16^{ \log_{16}5}= \boxed{5}$

In proper $LaTex$ presentation:

$\LARGE 4^{\log_{16} 25} = 4^{\frac{\log_{4} 25}{\log_{4} 16}} = 4^{\frac{\log_{4} 25}{2}} = 4^{\log_{4} \sqrt{25}} = 4^{\log_{4} 5} = \boxed{5}$

{ 4 }^{ \log _{ 16 }{ 25 } }={ 4 }^{ \frac { 1 }{ 2 } \log _{ 4 }{ 25 } }={ 4 }^{ \log _{ 4 }{ { 25 }^{ \frac { 1 }{ 2 } } } }={ 4 }^{ \log _{ 4 }{ 5 } }=\boxed { 5 }

$\Large {4}^{\log_{16}{25}}$

$\Large = {16}^{\frac{1}{2}\log_{16}{25}}$

$\Large = {16}^{\log_{16}{5}}$

$\Large = \color{#20A900}{\boxed{5}}$

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Note :

$\Large \color{#20A900}{a} \log_{x}{\color{#3D99F6}{b}}=\log_{x}{\color{#3D99F6}{b}}^{\color{#20A900}{a}}$

$\Large \color{#20A900}{a}^{\log_{\color{#20A900}{a}}{\color{#3D99F6}{b}}}=\color{#3D99F6}{b}$