Precalculus Examples

$\log_{6} (\sqrt[9]{\frac{s^{6} t}{36}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{\frac{s^{6} t}{36}}$ as ${(\frac{s^{6} t}{36})}^{\frac{1}{9}}$ .

$\log_{6} ({(\frac{s^{6} t}{36})}^{\frac{1}{9}})$

Step 2

Expand $\log_{6} ({(\frac{s^{6} t}{36})}^{\frac{1}{9}})$ by moving $\frac{1}{9}$ outside the logarithm.

$\frac{1}{9} \log_{6} (\frac{s^{6} t}{36})$

Step 3

Rewrite $\log_{6} (\frac{s^{6} t}{36})$ as $\log_{6} (s^{6} t) - \log_{6} (36)$ .

$\frac{1}{9} (\log_{6} (s^{6} t) - \log_{6} (36))$

Step 4

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

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

Step 5

Multiply $- 1$ by $2$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

Cancel the common factor of $3$ .

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Step 9.1.1

Factor $3$ out of $9$ .

$\frac{1}{3 (3)} (6 \log_{6} (s)) + \frac{1}{9} \log_{6} (t) + \frac{1}{9} \cdot - 2$

Step 9.1.2

Factor $3$ out of $6 \log_{6} (s)$ .

$\frac{1}{3 (3)} (3 (2 \log_{6} (s))) + \frac{1}{9} \log_{6} (t) + \frac{1}{9} \cdot - 2$

Step 9.1.3

Cancel the common factor.

$\frac{1}{3 \cdot 3} (3 (2 \log_{6} (s))) + \frac{1}{9} \log_{6} (t) + \frac{1}{9} \cdot - 2$

Step 9.1.4

Rewrite the expression.

$\frac{1}{3} (2 \log_{6} (s)) + \frac{1}{9} \log_{6} (t) + \frac{1}{9} \cdot - 2$

Step 9.2

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

$\frac{2}{3} \log_{6} (s) + \frac{1}{9} \log_{6} (t) + \frac{1}{9} \cdot - 2$

Step 9.3

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

$\frac{2 \log_{6} (s)}{3} + \frac{1}{9} \log_{6} (t) + \frac{1}{9} \cdot - 2$

Step 9.4

Combine $\frac{1}{9}$ and $\log_{6} (t)$ .

$\frac{2 \log_{6} (s)}{3} + \frac{\log_{6} (t)}{9} + \frac{1}{9} \cdot - 2$

Step 9.5

Combine $\frac{1}{9}$ and $- 2$ .

$\frac{2 \log_{6} (s)}{3} + \frac{\log_{6} (t)}{9} + \frac{- 2}{9}$

Step 10

Move the negative in front of the fraction.

$\frac{2 \log_{6} (s)}{3} + \frac{\log_{6} (t)}{9} - \frac{2}{9}$