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

$\ln (\sqrt{\frac{x}{y}})$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

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

Step 6

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

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