Question

Use the properties of logarithms to expand the following expression!

Answer 1

ANSWER

$\frac{7}{2}logy-logx-3logz$

EXPLANATION

Given;

$\log \left(\frac{\sqrt{y^7}\:}{xz^3\:}\right)$

Rewrite ;

$log\sqrt{y^7}-log(xz^3)$

Apply log rule;

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

Hence;

Apply log rule;

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