Precalculus Examples

$\log (\sqrt[5]{\frac{x}{y}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{x}{y}}$ as ${(\frac{x}{y})}^{\frac{1}{5}}$ .

$\log ({(\frac{x}{y})}^{\frac{1}{5}})$

Step 2

Expand $\log ({(\frac{x}{y})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log (\frac{x}{y})$

Step 3

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

$\frac{1}{5} (\log (x) - \log (y))$

Step 4

Apply the distributive property.

$\frac{1}{5} \log (x) + \frac{1}{5} (- \log (y))$

Step 5

Combine $\frac{1}{5}$ and $\log (x)$ .

$\frac{\log (x)}{5} + \frac{1}{5} (- \log (y))$

Step 6

Combine $\frac{1}{5}$ and $\log (y)$ .

$\frac{\log (x)}{5} - \frac{\log (y)}{5}$