Precalculus Examples

$\log_{3} (\sqrt[5]{\frac{x^{4} y}{9}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{x^{4} y}{9}}$ as ${(\frac{x^{4} y}{9})}^{\frac{1}{5}}$ .

$\log_{3} ({(\frac{x^{4} y}{9})}^{\frac{1}{5}})$

Step 2

Expand $\log_{3} ({(\frac{x^{4} y}{9})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log_{3} (\frac{x^{4} y}{9})$

Step 3

Rewrite $\log_{3} (\frac{x^{4} y}{9})$ as $\log_{3} (x^{4} y) - \log_{3} (9)$ .

$\frac{1}{5} (\log_{3} (x^{4} y) - \log_{3} (9))$

Step 4

Logarithm base $3$ of $9$ is $2$ .

$\frac{1}{5} (\log_{3} (x^{4} y) - 1 \cdot 2)$

Step 5

Multiply $- 1$ by $2$ .

$\frac{1}{5} (\log_{3} (x^{4} y) - 2)$

Step 6

Rewrite $\log_{3} (x^{4} y)$ as $\log_{3} (x^{4}) + \log_{3} (y)$ .

$\frac{1}{5} (\log_{3} (x^{4}) + \log_{3} (y) - 2)$

Step 7

Expand $\log_{3} (x^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{5} (4 \log_{3} (x) + \log_{3} (y) - 2)$

Step 8

Apply the distributive property.

$\frac{1}{5} (4 \log_{3} (x)) + \frac{1}{5} \log_{3} (y) + \frac{1}{5} \cdot - 2$

Step 9

Simplify.

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Step 9.1

Multiply $\frac{1}{5} (4 \log_{3} (x))$ .

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Step 9.1.1

Combine $4$ and $\frac{1}{5}$ .

$\frac{4}{5} \log_{3} (x) + \frac{1}{5} \log_{3} (y) + \frac{1}{5} \cdot - 2$

Step 9.1.2

Combine $\frac{4}{5}$ and $\log_{3} (x)$ .

$\frac{4 \log_{3} (x)}{5} + \frac{1}{5} \log_{3} (y) + \frac{1}{5} \cdot - 2$

Step 9.2

Combine $\frac{1}{5}$ and $\log_{3} (y)$ .

$\frac{4 \log_{3} (x)}{5} + \frac{\log_{3} (y)}{5} + \frac{1}{5} \cdot - 2$

Step 9.3

Combine $\frac{1}{5}$ and $- 2$ .

$\frac{4 \log_{3} (x)}{5} + \frac{\log_{3} (y)}{5} + \frac{- 2}{5}$

Step 10

Move the negative in front of the fraction.

$\frac{4 \log_{3} (x)}{5} + \frac{\log_{3} (y)}{5} - \frac{2}{5}$