Question 185449
*[Tex \LARGE \log_{8}\left(64\right)] ... Start with the given expression.



*[Tex \LARGE y=\log_{8}\left(64\right)] ... Set "y" equal to the expression



*[Tex \LARGE 8^{y}=64] ... Rewrite the equation using the property: *[Tex \LARGE \log_{b}\left(x\right)=b^y=x\Leftrightarrow y]


You can probably see that the answer is 2 (since 8 squared is 64), but let's keep going...



*[Tex \LARGE \left(2^3\right)^{y}=64] ... Rewrite {{{8}}} as {{{2^3}}}



*[Tex \LARGE \left(2^3\right)^{y}=2^6] ... Rewrite {{{64}}} as {{{2^6}}}



*[Tex \LARGE 2^{3y}=2^6] ... Multiply the exponents.



Since the bases are both equal to 2, this means that the exponents are equal.



*[Tex \LARGE 3y=6] ... Set the exponents equal to one another.



*[Tex \LARGE y=\frac{6}{3}] ... Divide both sides by 3.



*[Tex \LARGE y=2] ... Reduce




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Answer:



So the solution is *[Tex \LARGE y=2]



However, remember that we let *[Tex \LARGE y=\log_{8}\left(64\right)] (in the second step), to this means that 


*[Tex \LARGE \log_{8}\left(64\right)=2]