Precalculus Examples

$\log_{6} (\sqrt[3]{\frac{x^{2} y}{36}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{x^{2} y}{36}}$ as ${(\frac{x^{2} y}{36})}^{\frac{1}{3}}$ .

$\log_{6} ({(\frac{x^{2} y}{36})}^{\frac{1}{3}})$

Step 2

Expand $\log_{6} ({(\frac{x^{2} y}{36})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{6} (\frac{x^{2} y}{36})$

Step 3

Rewrite $\log_{6} (\frac{x^{2} y}{36})$ as $\log_{6} (x^{2} y) - \log_{6} (36)$ .

$\frac{1}{3} (\log_{6} (x^{2} y) - \log_{6} (36))$

Step 4

Logarithm base $6$ of $36$ is $2$ .

$\frac{1}{3} (\log_{6} (x^{2} y) - 1 \cdot 2)$

Step 5

Multiply $- 1$ by $2$ .

$\frac{1}{3} (\log_{6} (x^{2} y) - 2)$

Step 6

Rewrite $\log_{6} (x^{2} y)$ as $\log_{6} (x^{2}) + \log_{6} (y)$ .

$\frac{1}{3} (\log_{6} (x^{2}) + \log_{6} (y) - 2)$

Step 7

Expand $\log_{6} (x^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{3} (2 \log_{6} (x) + \log_{6} (y) - 2)$

Step 8

Apply the distributive property.

$\frac{1}{3} (2 \log_{6} (x)) + \frac{1}{3} \log_{6} (y) + \frac{1}{3} \cdot - 2$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Multiply $\frac{1}{3} (2 \log_{6} (x))$ .

Tap for more steps...

Step 9.1.1

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} \log_{6} (x) + \frac{1}{3} \log_{6} (y) + \frac{1}{3} \cdot - 2$

Step 9.1.2

Combine $\frac{2}{3}$ and $\log_{6} (x)$ .

$\frac{2 \log_{6} (x)}{3} + \frac{1}{3} \log_{6} (y) + \frac{1}{3} \cdot - 2$

Step 9.2

Combine $\frac{1}{3}$ and $\log_{6} (y)$ .

$\frac{2 \log_{6} (x)}{3} + \frac{\log_{6} (y)}{3} + \frac{1}{3} \cdot - 2$

Step 9.3

Combine $\frac{1}{3}$ and $- 2$ .

$\frac{2 \log_{6} (x)}{3} + \frac{\log_{6} (y)}{3} + \frac{- 2}{3}$

Step 10

Move the negative in front of the fraction.

$\frac{2 \log_{6} (x)}{3} + \frac{\log_{6} (y)}{3} - \frac{2}{3}$