Precalculus Examples

$\log_{6} (\sqrt[6]{\frac{x^{7}}{y^{6} z^{7}}})$

Step 1

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

$\log_{6} ({(\frac{x^{7}}{y^{6} z^{7}})}^{\frac{1}{6}})$

Step 2

Expand $\log_{6} ({(\frac{x^{7}}{y^{6} z^{7}})}^{\frac{1}{6}})$ by moving $\frac{1}{6}$ outside the logarithm.

$\frac{1}{6} \log_{6} (\frac{x^{7}}{y^{6} z^{7}})$

Step 3

Rewrite $\log_{6} (\frac{x^{7}}{y^{6} z^{7}})$ as $\log_{6} (x^{7}) - \log_{6} (y^{6} z^{7})$ .

$\frac{1}{6} (\log_{6} (x^{7}) - \log_{6} (y^{6} z^{7}))$

Step 4

Expand $\log_{6} (x^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{6} (7 \log_{6} (x) - \log_{6} (y^{6} z^{7}))$

Step 5

Rewrite $\log_{6} (y^{6} z^{7})$ as $\log_{6} (y^{6}) + \log_{6} (z^{7})$ .

$\frac{1}{6} (7 \log_{6} (x) - (\log_{6} (y^{6}) + \log_{6} (z^{7})))$

Step 6

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

$\frac{1}{6} (7 \log_{6} (x) - (6 \log_{6} (y) + \log_{6} (z^{7})))$

Step 7

Expand $\log_{6} (z^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{6} (7 \log_{6} (x) - (6 \log_{6} (y) + 7 \log_{6} (z)))$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{6} (7 \log_{6} (x) - (6 \log_{6} (y)) - (7 \log_{6} (z)))$

Step 8.2

Multiply $6$ by $- 1$ .

$\frac{1}{6} (7 \log_{6} (x) - 6 \log_{6} (y) - (7 \log_{6} (z)))$

Step 8.3

Multiply $7$ by $- 1$ .

$\frac{1}{6} (7 \log_{6} (x) - 6 \log_{6} (y) - 7 \log_{6} (z))$

Step 9

Apply the distributive property.

$\frac{1}{6} (7 \log_{6} (x)) + \frac{1}{6} (- 6 \log_{6} (y)) + \frac{1}{6} (- 7 \log_{6} (z))$

Step 10

Simplify.

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Step 10.1

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

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Step 10.1.1

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

$\frac{7}{6} \log_{6} (x) + \frac{1}{6} (- 6 \log_{6} (y)) + \frac{1}{6} (- 7 \log_{6} (z))$

Step 10.1.2

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

$\frac{7 \log_{6} (x)}{6} + \frac{1}{6} (- 6 \log_{6} (y)) + \frac{1}{6} (- 7 \log_{6} (z))$

Step 10.2

Cancel the common factor of $6$ .

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Step 10.2.1

Factor $6$ out of $- 6 \log_{6} (y)$ .

$\frac{7 \log_{6} (x)}{6} + \frac{1}{6} (6 (- \log_{6} (y))) + \frac{1}{6} (- 7 \log_{6} (z))$

Step 10.2.2

Cancel the common factor.

$\frac{7 \log_{6} (x)}{6} + \frac{1}{6} (6 (- \log_{6} (y))) + \frac{1}{6} (- 7 \log_{6} (z))$

Step 10.2.3

Rewrite the expression.

$\frac{7 \log_{6} (x)}{6} - \log_{6} (y) + \frac{1}{6} (- 7 \log_{6} (z))$

Step 10.3

Multiply $\frac{1}{6} (- 7 \log_{6} (z))$ .

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Step 10.3.1

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

$\frac{7 \log_{6} (x)}{6} - \log_{6} (y) + \frac{- 7}{6} \log_{6} (z)$

Step 10.3.2

Combine $\frac{- 7}{6}$ and $\log_{6} (z)$ .

$\frac{7 \log_{6} (x)}{6} - \log_{6} (y) + \frac{- 7 \log_{6} (z)}{6}$

Step 11

Move the negative in front of the fraction.

$\frac{7 \log_{6} (x)}{6} - \log_{6} (y) - \frac{7 \log_{6} (z)}{6}$