Answer
See the explanation
Work Step by Step
Law of Logarithms:
1. $\log(AB)=\log A +\log B$
2. $\log(A/B)=\log A -\log B$
3. $\log(A^C)=C\log A$
4. $\log_{A}B=\frac{\log_{C}A}{\log_{C}B}$
Properties of Loagrithms:
1. $\log_{a}1=0$
2. $\log_{a}a=1$
3. $\log_{a}a^x=x$
4. $a^{\log_{a}x}=x$
a. From law of logarithms 2 and then law of logarithms 1...
$\log (\frac{xy^3}{z^2})=\log(xy^3)-\log z^2=\log x+3\log y-2\log z$
b. From law of logarithms 3 and then law of logarithms 2...
$\ln \sqrt{\frac{x}{y}}=\frac{1}{2} \log{\frac{x}{y}}=\frac{1}{2}(\log x-\log y)$
c. From law of logarithms 3 and then law of logarithms 2 and then law of logarithms 1...
$\log {\sqrt[3] {\frac{x+2}{x^4(x^2+4)}}}=\frac{1}{3}\log\left(\frac{x+2}{x^4(x^2+4)}\right)=\frac{1}{3}\left(\log(x+2)-4\log x-\log(x^2+4)\right)$
