Precalculus Examples

$\log_{9} (\sqrt[5]{\frac{x^{2 y}}{81}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{x^{2 y}}{81}}$ as ${(\frac{x^{2 y}}{81})}^{\frac{1}{5}}$ .

$\log_{9} ({(\frac{x^{2 y}}{81})}^{\frac{1}{5}})$

Step 2

Expand $\log_{9} ({(\frac{x^{2 y}}{81})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log_{9} (\frac{x^{2 y}}{81})$

Step 3

Rewrite $\log_{9} (\frac{x^{2 y}}{81})$ as $\log_{9} (x^{2 y}) - \log_{9} (81)$ .

$\frac{1}{5} (\log_{9} (x^{2 y}) - \log_{9} (81))$

Step 4

Expand $\log_{9} (x^{2 y})$ by moving $2 y$ outside the logarithm.

$\frac{1}{5} (2 y \log_{9} (x) - \log_{9} (81))$

Step 5

Logarithm base $9$ of $81$ is $2$ .

$\frac{1}{5} (2 y \log_{9} (x) - 1 \cdot 2)$

Step 6

Multiply $- 1$ by $2$ .

$\frac{1}{5} (2 y \log_{9} (x) - 2)$

Step 7

Apply the distributive property.

$\frac{1}{5} (2 y \log_{9} (x)) + \frac{1}{5} \cdot - 2$

Step 8

Multiply $\frac{1}{5} (2 y \log_{9} (x))$ .

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Step 8.1

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

$\frac{2}{5} (y \log_{9} (x)) + \frac{1}{5} \cdot - 2$

Step 8.2

Combine $y$ and $\frac{2}{5}$ .

$\frac{y \cdot 2}{5} \log_{9} (x) + \frac{1}{5} \cdot - 2$

Step 8.3

Combine $\frac{y \cdot 2}{5}$ and $\log_{9} (x)$ .

$\frac{y \cdot 2 \log_{9} (x)}{5} + \frac{1}{5} \cdot - 2$

Step 9

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

$\frac{y \cdot 2 \log_{9} (x)}{5} + \frac{- 2}{5}$

Step 10

Simplify each term.

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Step 10.1

Move $2$ to the left of $y$ .

$\frac{2 \cdot y \log_{9} (x)}{5} + \frac{- 2}{5}$

Step 10.2

Move the negative in front of the fraction.

$\frac{2 y \log_{9} (x)}{5} - \frac{2}{5}$