Expand the Logarithmic Expression natural log of y square root of y/(1-y)

$\ln (y \sqrt{\frac{y}{1 - y}})$

Step 1

Rewrite $\ln (y \sqrt{\frac{y}{1 - y}})$ as $\ln (y) + \ln (\sqrt{\frac{y}{1 - y}})$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\ln (\frac{y}{1 - y})$ as $\ln (y) - \ln (1 - y)$ .

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

Step 5

Simplify each term.

Apply the distributive property.

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

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

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

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

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

Combine the numerators over the common denominator.

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