Question 197092
Remember, *[Tex \LARGE b^{\log_{b}\left(x\right)}=x] is one identity that is useful for simplifying logarithms.



So *[Tex \LARGE 2^{\log_{2}\left(5\right)}=5] (in this case, {{{b=2}}} and {{{x=5}}})



This means that 



*[Tex \LARGE 4^{\log_{2}\left(2^{\log_{2}\left(5\right)}\right)}]



simplifies to 



*[Tex \LARGE 4^{\log_{2}\left(5\right)}]



In other words, *[Tex \LARGE 4^{\log_{2}\left(2^{\log_{2}\left(5\right)}\right)}=4^{\log_{2}\left(5\right)}]



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Now let's simplify *[Tex \LARGE 4^{\log_{2}\left(5\right)}]




*[Tex \LARGE \left(2^2\right)^{\log_{2}\left(5\right)}] ... Rewrite {{{4}}} as {{{2^2}}}



*[Tex \LARGE \left(2\right)^{2\log_{2}\left(5\right)}] ... Multiply the exponents.



*[Tex \LARGE \left(2\right)^{\log_{2}\left(5^2\right)}] ... Rewrite the inner log using the identity  {{{y*log(b,(x))=log(b,(x^y))}}}



*[Tex \LARGE \left(2\right)^{\log_{2}\left(25\right)}] ... Square 5 to get 25



*[Tex \LARGE 25] ... Use the first identity given to simplify



So *[Tex \LARGE 4^{\log_{2}\left(5\right)}=25]



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Answer:



This means that *[Tex \LARGE 4^{\log_{2}\left(2^{\log_{2}\left(5\right)}\right)}=25]