Precalculus Examples

$\log_{9} (\sqrt[7]{\frac{s^{6} t}{81}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{\frac{s^{6} t}{81}}$ as ${(\frac{s^{6} t}{81})}^{\frac{1}{7}}$ .

$\log_{9} ({(\frac{s^{6} t}{81})}^{\frac{1}{7}})$

Step 2

Expand $\log_{9} ({(\frac{s^{6} t}{81})}^{\frac{1}{7}})$ by moving $\frac{1}{7}$ outside the logarithm.

$\frac{1}{7} \log_{9} (\frac{s^{6} t}{81})$

Step 3

Rewrite $\log_{9} (\frac{s^{6} t}{81})$ as $\log_{9} (s^{6} t) - \log_{9} (81)$ .

$\frac{1}{7} (\log_{9} (s^{6} t) - \log_{9} (81))$

Step 4

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

$\frac{1}{7} (\log_{9} (s^{6} t) - 1 \cdot 2)$

Step 5

Multiply $- 1$ by $2$ .

$\frac{1}{7} (\log_{9} (s^{6} t) - 2)$

Step 6

Rewrite $\log_{9} (s^{6} t)$ as $\log_{9} (s^{6}) + \log_{9} (t)$ .

$\frac{1}{7} (\log_{9} (s^{6}) + \log_{9} (t) - 2)$

Step 7

Expand $\log_{9} (s^{6})$ by moving $6$ outside the logarithm.

$\frac{1}{7} (6 \log_{9} (s) + \log_{9} (t) - 2)$

Step 8

Apply the distributive property.

$\frac{1}{7} (6 \log_{9} (s)) + \frac{1}{7} \log_{9} (t) + \frac{1}{7} \cdot - 2$

Step 9

Simplify.

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Step 9.1

Multiply $\frac{1}{7} (6 \log_{9} (s))$ .

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Step 9.1.1

Combine $6$ and $\frac{1}{7}$ .

$\frac{6}{7} \log_{9} (s) + \frac{1}{7} \log_{9} (t) + \frac{1}{7} \cdot - 2$

Step 9.1.2

Combine $\frac{6}{7}$ and $\log_{9} (s)$ .

$\frac{6 \log_{9} (s)}{7} + \frac{1}{7} \log_{9} (t) + \frac{1}{7} \cdot - 2$

Step 9.2

Combine $\frac{1}{7}$ and $\log_{9} (t)$ .

$\frac{6 \log_{9} (s)}{7} + \frac{\log_{9} (t)}{7} + \frac{1}{7} \cdot - 2$

Step 9.3

Combine $\frac{1}{7}$ and $- 2$ .

$\frac{6 \log_{9} (s)}{7} + \frac{\log_{9} (t)}{7} + \frac{- 2}{7}$

Step 10

Move the negative in front of the fraction.

$\frac{6 \log_{9} (s)}{7} + \frac{\log_{9} (t)}{7} - \frac{2}{7}$