Precalculus Examples

$\log_{b} (\sqrt[6]{\frac{x^{5}}{y^{6} z^{5}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{\frac{x^{5}}{y^{6} z^{5}}}$ as ${(\frac{x^{5}}{y^{6} z^{5}})}^{\frac{1}{6}}$ .

$\log_{b} ({(\frac{x^{5}}{y^{6} z^{5}})}^{\frac{1}{6}})$

Step 2

Expand $\log_{b} ({(\frac{x^{5}}{y^{6} z^{5}})}^{\frac{1}{6}})$ by moving $\frac{1}{6}$ outside the logarithm.

$\frac{1}{6} \log_{b} (\frac{x^{5}}{y^{6} z^{5}})$

Step 3

Rewrite $\log_{b} (\frac{x^{5}}{y^{6} z^{5}})$ as $\log_{b} (x^{5}) - \log_{b} (y^{6} z^{5})$ .

$\frac{1}{6} (\log_{b} (x^{5}) - \log_{b} (y^{6} z^{5}))$

Step 4

Expand $\log_{b} (x^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{6} (5 \log_{b} (x) - \log_{b} (y^{6} z^{5}))$

Step 5

Rewrite $\log_{b} (y^{6} z^{5})$ as $\log_{b} (y^{6}) + \log_{b} (z^{5})$ .

$\frac{1}{6} (5 \log_{b} (x) - (\log_{b} (y^{6}) + \log_{b} (z^{5})))$

Step 6

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

$\frac{1}{6} (5 \log_{b} (x) - (6 \log_{b} (y) + \log_{b} (z^{5})))$

Step 7

Expand $\log_{b} (z^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{6} (5 \log_{b} (x) - (6 \log_{b} (y) + 5 \log_{b} (z)))$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{6} (5 \log_{b} (x) - (6 \log_{b} (y)) - (5 \log_{b} (z)))$

Step 8.2

Multiply $6$ by $- 1$ .

$\frac{1}{6} (5 \log_{b} (x) - 6 \log_{b} (y) - (5 \log_{b} (z)))$

Step 8.3

Multiply $5$ by $- 1$ .

$\frac{1}{6} (5 \log_{b} (x) - 6 \log_{b} (y) - 5 \log_{b} (z))$

Step 9

Apply the distributive property.

$\frac{1}{6} (5 \log_{b} (x)) + \frac{1}{6} (- 6 \log_{b} (y)) + \frac{1}{6} (- 5 \log_{b} (z))$

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{6} (5 \log_{b} (x))$ .

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Step 10.1.1

Combine $5$ and $\frac{1}{6}$ .

$\frac{5}{6} \log_{b} (x) + \frac{1}{6} (- 6 \log_{b} (y)) + \frac{1}{6} (- 5 \log_{b} (z))$

Step 10.1.2

Combine $\frac{5}{6}$ and $\log_{b} (x)$ .

$\frac{5 \log_{b} (x)}{6} + \frac{1}{6} (- 6 \log_{b} (y)) + \frac{1}{6} (- 5 \log_{b} (z))$

Step 10.2

Cancel the common factor of $6$ .

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Step 10.2.1

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

$\frac{5 \log_{b} (x)}{6} + \frac{1}{6} (6 (- \log_{b} (y))) + \frac{1}{6} (- 5 \log_{b} (z))$

Step 10.2.2

Cancel the common factor.

$\frac{5 \log_{b} (x)}{6} + \frac{1}{6} (6 (- \log_{b} (y))) + \frac{1}{6} (- 5 \log_{b} (z))$

Step 10.2.3

Rewrite the expression.

$\frac{5 \log_{b} (x)}{6} - \log_{b} (y) + \frac{1}{6} (- 5 \log_{b} (z))$

Step 10.3

Multiply $\frac{1}{6} (- 5 \log_{b} (z))$ .

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Step 10.3.1

Combine $- 5$ and $\frac{1}{6}$ .

$\frac{5 \log_{b} (x)}{6} - \log_{b} (y) + \frac{- 5}{6} \log_{b} (z)$

Step 10.3.2

Combine $\frac{- 5}{6}$ and $\log_{b} (z)$ .

$\frac{5 \log_{b} (x)}{6} - \log_{b} (y) + \frac{- 5 \log_{b} (z)}{6}$

Step 11

Move the negative in front of the fraction.

$\frac{5 \log_{b} (x)}{6} - \log_{b} (y) - \frac{5 \log_{b} (z)}{6}$