Question

Use the properties of logarithms to expand the following expression!

Answer 1

`(7)/(2)logy-logx-3logz`

Step-by-step explanation

Given;

`\log \left((√(y^7)\:)/(xz^3\:)\right)`

Rewrite ;

`log√(y^7)-log(xz^3)`

Apply log rule;

`\begin{gathered} \log _c\left(ab\right)=\log _c\left(a\right)+\log _c\left(b\right) \\ logy^(7)-log(xz^(3)) \\ =\log_(10)\left(√(y^7)\right)-log\left(x\right)+log(z^3) \\ \end{gathered}`

Hence;

Apply log rule;

`\begin{gathered} \log _a\left(x^b\right)=b\cdot \log _a\left(x\right) \\ \log _(10)\left(z^3\right)=3\log _(10)\left(z\right) \\ =\log_(10)\left(√(y^7)\right)-\left(\log_(10)\left(x\right)+3\log_(10)\left(z\right)\right) \\ \log_(10)\left(√(y^7)\right)=(7)/(2)logy \\ \Rightarrow(7)/(2)logy-log\left(x\right)-3\log_(z) \\ (7)/(2)logy- logx-3logz \end{gathered}`