Answer
a) $ \log x + 3\log y - 2\log z $
b) $\frac{1}{2} \ln x - \frac{1}{2} \ln y$
c) $ \frac{1}{3} \log (x+2) - \frac{4}{3}\log x-\frac{1}{3}\log(x^{2}+4)$
Work Step by Step
Use the basic Laws of Logarithms to expand the espressions:
a) $\log(\frac{xy^{3}}{z^{2}})= \log x + 3\log y - 2\log z $
b) $\ln\sqrt \frac{x}{y}=\frac{1}{2} \ln x - \frac{1}{2} \ln y$
c) $ \log\sqrt[3] \frac{x+2}{x^{4}(x^{2}+4)}=\log\sqrt[3] {x+2} - \log \sqrt[3] {x^{4}(x^{2}+4)}= \frac{1}{3} \log (x+2) - \frac{1}{3}\log (x^{4}(x^{2}+4))=\frac{1}{3} \log (x+2) - [\frac{1}{3}\log x^{4}+\frac{1}{3}\log(x^{2}+4)]=\frac{1}{3} \log (x+2) - \frac{4}{3}\log x-\frac{1}{3}\log(x^{2}+4)$