Precalculus Examples

$\log_{b} (\sqrt[3]{\frac{x^{5} y^{8}}{z^{2}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{x^{5} y^{8}}{z^{2}}}$ as ${(\frac{x^{5} y^{8}}{z^{2}})}^{\frac{1}{3}}$ .

$\log_{b} ({(\frac{x^{5} y^{8}}{z^{2}})}^{\frac{1}{3}})$

Step 2

Expand $\log_{b} ({(\frac{x^{5} y^{8}}{z^{2}})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{b} (\frac{x^{5} y^{8}}{z^{2}})$

Step 3

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

$\frac{1}{3} (\log_{b} (x^{5} y^{8}) - \log_{b} (z^{2}))$

Step 4

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

$\frac{1}{3} (\log_{b} (x^{5} y^{8}) - (2 \log_{b} (z)))$

Step 5

Multiply $2$ by $- 1$ .

$\frac{1}{3} (\log_{b} (x^{5} y^{8}) - 2 \log_{b} (z))$

Step 6

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

$\frac{1}{3} (\log_{b} (x^{5}) + \log_{b} (y^{8}) - 2 \log_{b} (z))$

Step 7

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

$\frac{1}{3} (5 \log_{b} (x) + \log_{b} (y^{8}) - 2 \log_{b} (z))$

Step 8

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

$\frac{1}{3} (5 \log_{b} (x) + 8 \log_{b} (y) - 2 \log_{b} (z))$

Step 9

Apply the distributive property.

$\frac{1}{3} (5 \log_{b} (x)) + \frac{1}{3} (8 \log_{b} (y)) + \frac{1}{3} (- 2 \log_{b} (z))$

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{3} (5 \log_{b} (x))$ .

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Step 10.1.1

Combine $5$ and $\frac{1}{3}$ .

$\frac{5}{3} \log_{b} (x) + \frac{1}{3} (8 \log_{b} (y)) + \frac{1}{3} (- 2 \log_{b} (z))$

Step 10.1.2

Combine $\frac{5}{3}$ and $\log_{b} (x)$ .

$\frac{5 \log_{b} (x)}{3} + \frac{1}{3} (8 \log_{b} (y)) + \frac{1}{3} (- 2 \log_{b} (z))$

Step 10.2

Multiply $\frac{1}{3} (8 \log_{b} (y))$ .

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Step 10.2.1

Combine $8$ and $\frac{1}{3}$ .

$\frac{5 \log_{b} (x)}{3} + \frac{8}{3} \log_{b} (y) + \frac{1}{3} (- 2 \log_{b} (z))$

Step 10.2.2

Combine $\frac{8}{3}$ and $\log_{b} (y)$ .

$\frac{5 \log_{b} (x)}{3} + \frac{8 \log_{b} (y)}{3} + \frac{1}{3} (- 2 \log_{b} (z))$

Step 10.3

Multiply $\frac{1}{3} (- 2 \log_{b} (z))$ .

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Step 10.3.1

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

$\frac{5 \log_{b} (x)}{3} + \frac{8 \log_{b} (y)}{3} + \frac{- 2}{3} \log_{b} (z)$

Step 10.3.2

Combine $\frac{- 2}{3}$ and $\log_{b} (z)$ .

$\frac{5 \log_{b} (x)}{3} + \frac{8 \log_{b} (y)}{3} + \frac{- 2 \log_{b} (z)}{3}$

Step 11

Move the negative in front of the fraction.

$\frac{5 \log_{b} (x)}{3} + \frac{8 \log_{b} (y)}{3} - \frac{2 \log_{b} (z)}{3}$