Expand the Logarithmic Expression log of ( square root of x)/( cube root of y)

$\log (\frac{\sqrt{x}}{\sqrt[3]{y}})$

Step 1

Rewrite $\log (\frac{\sqrt{x}}{\sqrt[3]{y}})$ as $\log (\sqrt{x}) - \log (\sqrt[3]{y})$ .

$\log (\sqrt{x}) - \log (\sqrt[3]{y})$

Step 2

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

$\log (x^{\frac{1}{2}}) - \log (\sqrt[3]{y})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

$\log (x^{\frac{1}{2}}) - \log (y^{\frac{1}{3}})$

Step 4

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

$\frac{1}{2} \log (x) - \log (y^{\frac{1}{3}})$

Step 5

Expand $\log (y^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{2} \log (x) - (\frac{1}{3} \log (y))$

Step 6

Combine $\frac{1}{3}$ and $\log (y)$ .

$\frac{1}{2} \log (x) - \frac{\log (y)}{3}$

Step 7

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

$\frac{\log (x)}{2} - \frac{\log (y)}{3}$