Question 221046
In order to evaluate *[Tex \LARGE 4^{\log_{8}(27)}], we need to simplify the exponent *[Tex \LARGE \log_{8}(27)]



*[Tex \LARGE \log_{8}(27)] ... Start with the given expression.



*[Tex \LARGE \frac{\log_{10}(27)}{\log_{10}(8)}] ... Use the change of base formula.



*[Tex \LARGE \frac{\log_{10}(3^3)}{\log_{10}(2^3)}] ... Rewrite 27 as {{{3^3}}} and rewrite 8 as {{{2^3}}}



*[Tex \LARGE \frac{3\log_{10}(3)}{3\log_{10}(2)}] ... Pull down the exponents using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



*[Tex \LARGE \frac{\log_{10}(3)}{\log_{10}(2)}] ... Cancel out the common terms.



*[Tex \LARGE \log_{2}(3)] ... Use the change of base formula.



So *[Tex \LARGE 4^{\log_{8}(27)}=4^{\log_{2}(3)}]



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*[Tex \LARGE 4^{\log_{2}(3)}] ... Start with the partially simplified expression.



*[Tex \LARGE (2^2)^{\log_{2}(3)}] ... Rewrite {{{4}}} as {{{2^2}}}



*[Tex \LARGE 2^{2\log_{2}(3)}] ... Multiply the exponents.



*[Tex \LARGE 2^{\log_{2}(3^2)}] ... Place the coefficient as the exponent using the identity  {{{y*log(b,(x))=log(b,(x^y))}}}



*[Tex \LARGE 3^2] ... Use the identity *[Tex \LARGE b^{\log_{b}(x)}=x] to simplify.



*[Tex \LARGE 9] ... Square 3 to get 9.



So *[Tex \LARGE 4^{\log_{2}(3)}=9]



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Answer:



This means that *[Tex \LARGE 4^{\log_{8}(27)}=9]