Precalculus Examples

$\log_{3} (\sqrt[5]{\sqrt{\frac{s^{8} t}{9}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{s^{8} t}{9}}$ as ${(\frac{s^{8} t}{9})}^{\frac{1}{2}}$ .

$\log_{3} (\sqrt[5]{{(\frac{s^{8} t}{9})}^{\frac{1}{2}}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(\frac{s^{8} t}{9})}^{\frac{1}{2}}}$ as ${(\frac{s^{8} t}{9})}^{\frac{\frac{1}{2}}{5}}$ .

$\log_{3} ({(\frac{s^{8} t}{9})}^{\frac{\frac{1}{2}}{5}})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log_{3} ({(\frac{s^{8} t}{9})}^{\frac{1}{2} \cdot \frac{1}{5}})$

Step 4

Multiply $\frac{1}{2} \cdot \frac{1}{5}$ .

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Step 4.1

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

$\log_{3} ({(\frac{s^{8} t}{9})}^{\frac{1}{2 \cdot 5}})$

Step 4.2

Multiply $2$ by $5$ .

$\log_{3} ({(\frac{s^{8} t}{9})}^{\frac{1}{10}})$

Step 5

Expand $\log_{3} ({(\frac{s^{8} t}{9})}^{\frac{1}{10}})$ by moving $\frac{1}{10}$ outside the logarithm.

$\frac{1}{10} \log_{3} (\frac{s^{8} t}{9})$

Step 6

Rewrite $\log_{3} (\frac{s^{8} t}{9})$ as $\log_{3} (s^{8} t) - \log_{3} (9)$ .

$\frac{1}{10} (\log_{3} (s^{8} t) - \log_{3} (9))$

Step 7

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

$\frac{1}{10} (\log_{3} (s^{8} t) - 1 \cdot 2)$

Step 8

Multiply $- 1$ by $2$ .

$\frac{1}{10} (\log_{3} (s^{8} t) - 2)$

Step 9

Rewrite $\log_{3} (s^{8} t)$ as $\log_{3} (s^{8}) + \log_{3} (t)$ .

$\frac{1}{10} (\log_{3} (s^{8}) + \log_{3} (t) - 2)$

Step 10

Expand $\log_{3} (s^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{10} (8 \log_{3} (s) + \log_{3} (t) - 2)$

Step 11

Apply the distributive property.

$\frac{1}{10} (8 \log_{3} (s)) + \frac{1}{10} \log_{3} (t) + \frac{1}{10} \cdot - 2$

Step 12

Simplify.

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Step 12.1

Cancel the common factor of $2$ .

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Step 12.1.1

Factor $2$ out of $10$ .

$\frac{1}{2 (5)} (8 \log_{3} (s)) + \frac{1}{10} \log_{3} (t) + \frac{1}{10} \cdot - 2$

Step 12.1.2

Factor $2$ out of $8 \log_{3} (s)$ .

$\frac{1}{2 (5)} (2 (4 \log_{3} (s))) + \frac{1}{10} \log_{3} (t) + \frac{1}{10} \cdot - 2$

Step 12.1.3

Cancel the common factor.

$\frac{1}{2 \cdot 5} (2 (4 \log_{3} (s))) + \frac{1}{10} \log_{3} (t) + \frac{1}{10} \cdot - 2$

Step 12.1.4

Rewrite the expression.

$\frac{1}{5} (4 \log_{3} (s)) + \frac{1}{10} \log_{3} (t) + \frac{1}{10} \cdot - 2$

Step 12.2

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

$\frac{4}{5} \log_{3} (s) + \frac{1}{10} \log_{3} (t) + \frac{1}{10} \cdot - 2$

Step 12.3

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

$\frac{4 \log_{3} (s)}{5} + \frac{1}{10} \log_{3} (t) + \frac{1}{10} \cdot - 2$

Step 12.4

Combine $\frac{1}{10}$ and $\log_{3} (t)$ .

$\frac{4 \log_{3} (s)}{5} + \frac{\log_{3} (t)}{10} + \frac{1}{10} \cdot - 2$

Step 12.5

Cancel the common factor of $2$ .

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Step 12.5.1

Factor $2$ out of $10$ .

$\frac{4 \log_{3} (s)}{5} + \frac{\log_{3} (t)}{10} + \frac{1}{2 (5)} \cdot - 2$

Step 12.5.2

Factor $2$ out of $- 2$ .

$\frac{4 \log_{3} (s)}{5} + \frac{\log_{3} (t)}{10} + \frac{1}{2 \cdot 5} \cdot (2 \cdot - 1)$

Step 12.5.3

Cancel the common factor.

$\frac{4 \log_{3} (s)}{5} + \frac{\log_{3} (t)}{10} + \frac{1}{2 \cdot 5} \cdot (2 \cdot - 1)$

Step 12.5.4

Rewrite the expression.

$\frac{4 \log_{3} (s)}{5} + \frac{\log_{3} (t)}{10} + \frac{1}{5} \cdot - 1$

Step 12.6

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

$\frac{4 \log_{3} (s)}{5} + \frac{\log_{3} (t)}{10} + \frac{- 1}{5}$

Step 13

Move the negative in front of the fraction.

$\frac{4 \log_{3} (s)}{5} + \frac{\log_{3} (t)}{10} - \frac{1}{5}$