Question 212356
*[Tex \LARGE \log_{b}\left( \sqrt[3]{\frac{x^8}{y^2z^5}} \right)] ... Start with the given expression.



*[Tex \LARGE \log_{b}\left( \left(\frac{x^8}{y^2z^5}\right)^{\frac{1}{3}} \right)] ... Convert to rational exponent notation.



*[Tex \LARGE \frac{1}{3}\log_{b}\left( \frac{x^8}{y^2z^5} \right)] ... Pull down the exponent using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



*[Tex \LARGE \frac{1}{3}\left(\log_{b}\left(x^8\right)-\log_{b}\left(y^2z^5\right) \right)] ... Break up the log using the identity  {{{log(b,(A/B))=log(b,(A))-log(b,(B))}}}



*[Tex \LARGE \frac{1}{3}\left(\log_{b}\left(x^8\right)-\left(\log_{b}\left(y^2\right)+\log_{b}\left(z^5\right)\right) \right)] ... Break up the second log using the identity  {{{log(b,(A*B))=log(b,(A))+log(b,(B))}}}



*[Tex \LARGE \frac{1}{3}\left(\log_{b}\left(x^8\right)-\log_{b}\left(y^2\right)-\log_{b}\left(z^5\right) \right)] Distribute



*[Tex \LARGE \frac{1}{3}\left(8\log_{b}\left(x\right)-2\log_{b}\left(y\right)-5\log_{b}\left(z\right) \right)] ... Pull down the exponents using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



*[Tex \LARGE \frac{8}{3}\log_{b}\left(x\right)-\frac{2}{3}\log_{b}\left(y\right)-\frac{5}{3}\log_{b}\left(z\right)] ... Distribute



So 


*[Tex \LARGE \log_{b}\left( \sqrt[3]{\frac{x^8}{y^2z^5}} \right) = \frac{8}{3}\log_{b}\left(x\right)-\frac{2}{3}\log_{b}\left(y\right)-\frac{5}{3}\log_{b}\left(z\right)] 



assuming that every variable is positive.