Expand the Logarithmic Expression log of cube root of x/y

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

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

Step 6

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

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