Finite Math Examples

$\log (\sqrt[5]{\frac{x^{4}}{y^{5} z^{2}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{x^{4}}{y^{5} z^{2}}}$ as ${(\frac{x^{4}}{y^{5} z^{2}})}^{\frac{1}{5}}$ .

$\log ({(\frac{x^{4}}{y^{5} z^{2}})}^{\frac{1}{5}})$

Step 2

Expand $\log ({(\frac{x^{4}}{y^{5} z^{2}})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log (\frac{x^{4}}{y^{5} z^{2}})$

Step 3

Rewrite $\log (\frac{x^{4}}{y^{5} z^{2}})$ as $\log (x^{4}) - \log (y^{5} z^{2})$ .

$\frac{1}{5} (\log (x^{4}) - \log (y^{5} z^{2}))$

Step 4

Expand $\log (x^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{5} (4 \log (x) - \log (y^{5} z^{2}))$

Step 5

Rewrite $\log (y^{5} z^{2})$ as $\log (y^{5}) + \log (z^{2})$ .

$\frac{1}{5} (4 \log (x) - (\log (y^{5}) + \log (z^{2})))$

Step 6

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

$\frac{1}{5} (4 \log (x) - (5 \log (y) + \log (z^{2})))$

Step 7

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

$\frac{1}{5} (4 \log (x) - (5 \log (y) + 2 \log (z)))$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{5} (4 \log (x) - (5 \log (y)) - (2 \log (z)))$

Step 8.2

Multiply $5$ by $- 1$ .

$\frac{1}{5} (4 \log (x) - 5 \log (y) - (2 \log (z)))$

Step 8.3

Multiply $2$ by $- 1$ .

$\frac{1}{5} (4 \log (x) - 5 \log (y) - 2 \log (z))$

Step 9

Apply the distributive property.

$\frac{1}{5} (4 \log (x)) + \frac{1}{5} (- 5 \log (y)) + \frac{1}{5} (- 2 \log (z))$

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{5} (4 \log (x))$ .

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Step 10.1.1

Combine $4$ and $\frac{1}{5}$ .

$\frac{4}{5} \log (x) + \frac{1}{5} (- 5 \log (y)) + \frac{1}{5} (- 2 \log (z))$

Step 10.1.2

Combine $\frac{4}{5}$ and $\log (x)$ .

$\frac{4 \log (x)}{5} + \frac{1}{5} (- 5 \log (y)) + \frac{1}{5} (- 2 \log (z))$

Step 10.2

Cancel the common factor of $5$ .

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Step 10.2.1

Factor $5$ out of $- 5 \log (y)$ .

$\frac{4 \log (x)}{5} + \frac{1}{5} (5 (- \log (y))) + \frac{1}{5} (- 2 \log (z))$

Step 10.2.2

Cancel the common factor.

$\frac{4 \log (x)}{5} + \frac{1}{5} (5 (- \log (y))) + \frac{1}{5} (- 2 \log (z))$

Step 10.2.3

Rewrite the expression.

$\frac{4 \log (x)}{5} - \log (y) + \frac{1}{5} (- 2 \log (z))$

Step 10.3

Multiply $\frac{1}{5} (- 2 \log (z))$ .

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Step 10.3.1

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

$\frac{4 \log (x)}{5} - \log (y) + \frac{- 2}{5} \log (z)$

Step 10.3.2

Combine $\frac{- 2}{5}$ and $\log (z)$ .

$\frac{4 \log (x)}{5} - \log (y) + \frac{- 2 \log (z)}{5}$

Step 11

Move the negative in front of the fraction.

$\frac{4 \log (x)}{5} - \log (y) - \frac{2 \log (z)}{5}$