Expand the Logarithmic Expression natural log of (x square root of y)/(zw^3)

$\ln (\frac{x \sqrt{y}}{z w^{3}})$

Step 1

Rewrite $\ln (\frac{x \sqrt{y}}{z w^{3}})$ as $\ln (x \sqrt{y}) - \ln (z w^{3})$ .

$\ln (x \sqrt{y}) - \ln (z w^{3})$

Step 2

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

$\ln (x y^{\frac{1}{2}}) - \ln (z w^{3})$

Step 3

Rewrite $\ln (x y^{\frac{1}{2}})$ as $\ln (x) + \ln (y^{\frac{1}{2}})$ .

$\ln (x) + \ln (y^{\frac{1}{2}}) - \ln (z w^{3})$

Step 4

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

$\ln (x) + \frac{1}{2} \ln (y) - \ln (z w^{3})$

Step 5

Rewrite $\ln (z w^{3})$ as $\ln (z) + \ln (w^{3})$ .

$\ln (x) + \frac{1}{2} \ln (y) - (\ln (z) + \ln (w^{3}))$

Step 6

Expand $\ln (w^{3})$ by moving $3$ outside the logarithm.

$\ln (x) + \frac{1}{2} \ln (y) - (\ln (z) + 3 \ln (w))$

Step 7

Simplify each term.

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

$\ln (x) + \frac{\ln (y)}{2} - (\ln (z) + 3 \ln (w))$

Apply the distributive property.

$\ln (x) + \frac{\ln (y)}{2} - \ln (z) - (3 \ln (w))$

Multiply $3$ by $- 1$ .

$\ln (x) + \frac{\ln (y)}{2} - \ln (z) - 3 \ln (w)$