Expand the Logarithmic Expression natural log of x^2 square root of y/z

$\ln (x^{2} \sqrt{\frac{y}{z}})$

Rewrite $\ln (x^{2} \sqrt{\frac{y}{z}})$ as $\ln (x^{2}) + \ln (\sqrt{\frac{y}{z}})$ .

$\ln (x^{2}) + \ln (\sqrt{\frac{y}{z}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{y}{z}}$ as ${(\frac{y}{z})}^{\frac{1}{2}}$ .

$\ln (x^{2}) + \ln ({(\frac{y}{z})}^{\frac{1}{2}})$

Expand $\ln (x^{2})$ by moving $2$ outside the logarithm.

$2 \ln (x) + \ln ({(\frac{y}{z})}^{\frac{1}{2}})$

Expand $\ln ({(\frac{y}{z})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Rewrite $\ln (\frac{y}{z})$ as $\ln (y) - \ln (z)$ .

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

Simplify each term.

Apply the distributive property.

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

Combine $\frac{1}{2}$ and $\ln (y)$ .

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

Combine $\frac{1}{2}$ and $\ln (z)$ .

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