Precalculus Examples

$\log_{b} (\sqrt[5]{\frac{x^{2}}{y^{5} z^{8}}})$

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

$\log_{b} ({(\frac{x^{2}}{y^{5} z^{8}})}^{\frac{1}{5}})$

Expand $\log_{b} ({(\frac{x^{2}}{y^{5} z^{8}})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log_{b} (\frac{x^{2}}{y^{5} z^{8}})$

Rewrite $\log_{b} (\frac{x^{2}}{y^{5} z^{8}})$ as $\log_{b} (x^{2}) - \log_{b} (y^{5} z^{8})$ .

$\frac{1}{5} (\log_{b} (x^{2}) - \log_{b} (y^{5} z^{8}))$

Expand $\log_{b} (x^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{5} (2 \log_{b} (x) - \log_{b} (y^{5} z^{8}))$

Rewrite $\log_{b} (y^{5} z^{8})$ as $\log_{b} (y^{5}) + \log_{b} (z^{8})$ .

$\frac{1}{5} (2 \log_{b} (x) - (\log_{b} (y^{5}) + \log_{b} (z^{8})))$

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

$\frac{1}{5} (2 \log_{b} (x) - (5 \log_{b} (y) + \log_{b} (z^{8})))$

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

$\frac{1}{5} (2 \log_{b} (x) - (5 \log_{b} (y) + 8 \log_{b} (z)))$

Simplify each term.

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Apply the distributive property.

$\frac{1}{5} (2 \log_{b} (x) - (5 \log_{b} (y)) - (8 \log_{b} (z)))$

Multiply $5$ by $- 1$ .

$\frac{1}{5} (2 \log_{b} (x) - 5 \log_{b} (y) - (8 \log_{b} (z)))$

Multiply $8$ by $- 1$ .

$\frac{1}{5} (2 \log_{b} (x) - 5 \log_{b} (y) - 8 \log_{b} (z))$

Apply the distributive property.

$\frac{1}{5} (2 \log_{b} (x)) + \frac{1}{5} (- 5 \log_{b} (y)) + \frac{1}{5} (- 8 \log_{b} (z))$

Simplify.

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Multiply $\frac{1}{5} (2 \log_{b} (x))$ .

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Combine $2$ and $\frac{1}{5}$ .

$\frac{2}{5} \log_{b} (x) + \frac{1}{5} (- 5 \log_{b} (y)) + \frac{1}{5} (- 8 \log_{b} (z))$

Combine $\frac{2}{5}$ and $\log_{b} (x)$ .

$\frac{2 \log_{b} (x)}{5} + \frac{1}{5} (- 5 \log_{b} (y)) + \frac{1}{5} (- 8 \log_{b} (z))$

Cancel the common factor of $5$ .

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Factor $5$ out of $- 5 \log_{b} (y)$ .

$\frac{2 \log_{b} (x)}{5} + \frac{1}{5} (5 (- \log_{b} (y))) + \frac{1}{5} (- 8 \log_{b} (z))$

Cancel the common factor.

$\frac{2 \log_{b} (x)}{5} + \frac{1}{5} (5 (- \log_{b} (y))) + \frac{1}{5} (- 8 \log_{b} (z))$

Rewrite the expression.

$\frac{2 \log_{b} (x)}{5} - \log_{b} (y) + \frac{1}{5} (- 8 \log_{b} (z))$

Multiply $\frac{1}{5} (- 8 \log_{b} (z))$ .

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Combine $- 8$ and $\frac{1}{5}$ .

$\frac{2 \log_{b} (x)}{5} - \log_{b} (y) + \frac{- 8}{5} \log_{b} (z)$

Combine $\frac{- 8}{5}$ and $\log_{b} (z)$ .

$\frac{2 \log_{b} (x)}{5} - \log_{b} (y) + \frac{- 8 \log_{b} (z)}{5}$

Move the negative in front of the fraction.

$\frac{2 \log_{b} (x)}{5} - \log_{b} (y) - \frac{8 \log_{b} (z)}{5}$