Precalculus Examples

$\ln (\sqrt[8]{e y})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{e y}$ as ${(e y)}^{\frac{1}{8}}$ .

$\ln ({(e y)}^{\frac{1}{8}})$

Step 2

Expand $\ln ({(e y)}^{\frac{1}{8}})$ by moving $\frac{1}{8}$ outside the logarithm.

$\frac{1}{8} \ln (e y)$

Step 3

Rewrite $\ln (e y)$ as $\ln (e) + \ln (y)$ .

$\frac{1}{8} (\ln (e) + \ln (y))$

Step 4

The natural logarithm of $e$ is $1$ .

$\frac{1}{8} (1 + \ln (y))$

Step 5

Apply the distributive property.

$\frac{1}{8} \cdot 1 + \frac{1}{8} \ln (y)$

Step 6

Multiply $\frac{1}{8}$ by $1$ .

$\frac{1}{8} + \frac{1}{8} \ln (y)$

Step 7

Combine $\frac{1}{8}$ and $\ln (y)$ .

$\frac{1}{8} + \frac{\ln (y)}{8}$