# Chapter 4 - Section 4.4 - Laws of Logarithms - 4.4 exercises: 43

$\ln x$ + $\frac{1}{2}$$($$\ln y$ - $\ln z$$) #### Work Step by Step Expand the expression: \ln (x\sqrt{\frac{y}{z}}) Apply the First Law of Logarithms \ln (x\times \sqrt{\frac{y}{z}}) = \ln x + \ln\sqrt{\frac{y}{z}} Rewrite the square root for \sqrt{\frac{y}{z}} \ln x + ln (\frac{y}{z})^\frac{1}{2} Apply the Third Law of Logarithms for ln (\frac{y}{z})^\frac{1}{2} ln (\frac{y}{z})^\frac{1}{2} = \frac{1}{2}$$\ln \frac{y}{z}$ Apply the Second Law of Logarithms for $\frac{1}{2}$$\ln \frac{y}{z} (Distribute the half) \frac{1}{2}$$\ln (\frac{y}{z})$ = $\frac{1}{2}$$\ln y - \frac{1}{2}$$\ln z$ Assemble the expression $\ln x$ + $\frac{1}{2}$$\ln y - \frac{1}{2}$$\ln z$ Factor out the $\frac{1}{2}$. We do this since one may put together the $\ln x$ and the $\frac{1}{2}$$\ln y when we combine it to one expression \ln x + \frac{1}{2}$$($$\ln y - \ln z$$)$

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