Chapter 4 - Exponential and Logarithmic Functions - Section 4.8 Exponential Growth and Decay Models; Newton's Law; Logistic Growth and Decay Models - 4.8 Assess Your Understanding - Page 359: 32

$2 \ln x+\frac{1}{2} \ln y-\ln z$

Use the rule $\ln \left(\frac{A}{B}\right)=\ln A-\ln B$ to obtain: $$\ln \left(\frac{x^{2} \sqrt{y}}{z}\right)=\ln \left(x^{2} \sqrt{y}\right)-\ln z$$ Use the rule $\ln (AB)=\ln A+\ln B$, and the fact that $\sqrt{x}=x^{\frac{1}{2}}$ to obtain: \begin{align*}\ln \left(x^{2} \sqrt{y}\right)-\ln z&=\left(\ln x^{2}+\ln \sqrt{y}\right)-\ln z\\ &=\ln x^{2}+\ln y^{1/2}-\ln z\\ \\\text{Use the rule }\ln a^{m}=m\cdot \ln{a} \text{ to obtain:}\\ \\\ln \left(x^{2} \sqrt{y}\right)-\ln z&=2 \ln x+\frac{1}{2} \ln y-\ln z \end{align*}