Calculus Examples

$\log_{4} (\sqrt[4]{\frac{x^{2} y^{10}}{z^{9}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{x^{2} y^{10}}{z^{9}}}$ as ${(\frac{x^{2} y^{10}}{z^{9}})}^{\frac{1}{4}}$ .

$\log_{4} ({(\frac{x^{2} y^{10}}{z^{9}})}^{\frac{1}{4}})$

Step 2

Expand $\log_{4} ({(\frac{x^{2} y^{10}}{z^{9}})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{4} (\frac{x^{2} y^{10}}{z^{9}})$

Step 3

Rewrite $\log_{4} (\frac{x^{2} y^{10}}{z^{9}})$ as $\log_{4} (x^{2} y^{10}) - \log_{4} (z^{9})$ .

$\frac{1}{4} (\log_{4} (x^{2} y^{10}) - \log_{4} (z^{9}))$

Step 4

Expand $\log_{4} (z^{9})$ by moving $9$ outside the logarithm.

$\frac{1}{4} (\log_{4} (x^{2} y^{10}) - (9 \log_{4} (z)))$

Step 5

Multiply $9$ by $- 1$ .

$\frac{1}{4} (\log_{4} (x^{2} y^{10}) - 9 \log_{4} (z))$

Step 6

Rewrite $\log_{4} (x^{2} y^{10})$ as $\log_{4} (x^{2}) + \log_{4} (y^{10})$ .

$\frac{1}{4} (\log_{4} (x^{2}) + \log_{4} (y^{10}) - 9 \log_{4} (z))$

Step 7

Expand $\log_{4} (x^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{4} (2 \log_{4} (x) + \log_{4} (y^{10}) - 9 \log_{4} (z))$

Step 8

Expand $\log_{4} (y^{10})$ by moving $10$ outside the logarithm.

$\frac{1}{4} (2 \log_{4} (x) + 10 \log_{4} (y) - 9 \log_{4} (z))$

Step 9

Apply the distributive property.

$\frac{1}{4} (2 \log_{4} (x)) + \frac{1}{4} (10 \log_{4} (y)) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10

Simplify.

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Step 10.1

Cancel the common factor of $2$ .

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Step 10.1.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (2 \log_{4} (x)) + \frac{1}{4} (10 \log_{4} (y)) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.1.2

Factor $2$ out of $2 \log_{4} (x)$ .

$\frac{1}{2 (2)} (2 (\log_{4} (x))) + \frac{1}{4} (10 \log_{4} (y)) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.1.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 \log_{4} (x)) + \frac{1}{4} (10 \log_{4} (y)) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.1.4

Rewrite the expression.

$\frac{1}{2} \log_{4} (x) + \frac{1}{4} (10 \log_{4} (y)) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.2

Combine $\frac{1}{2}$ and $\log_{4} (x)$ .

$\frac{\log_{4} (x)}{2} + \frac{1}{4} (10 \log_{4} (y)) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Factor $2$ out of $4$ .

$\frac{\log_{4} (x)}{2} + \frac{1}{2 (2)} (10 \log_{4} (y)) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.3.2

Factor $2$ out of $10 \log_{4} (y)$ .

$\frac{\log_{4} (x)}{2} + \frac{1}{2 (2)} (2 (5 \log_{4} (y))) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.3.3

Cancel the common factor.

$\frac{\log_{4} (x)}{2} + \frac{1}{2 \cdot 2} (2 (5 \log_{4} (y))) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.3.4

Rewrite the expression.

$\frac{\log_{4} (x)}{2} + \frac{1}{2} (5 \log_{4} (y)) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.4

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

$\frac{\log_{4} (x)}{2} + \frac{5}{2} \log_{4} (y) + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.5

Combine $\frac{5}{2}$ and $\log_{4} (y)$ .

$\frac{\log_{4} (x)}{2} + \frac{5 \log_{4} (y)}{2} + \frac{1}{4} (- 9 \log_{4} (z))$

Step 10.6

Multiply $\frac{1}{4} (- 9 \log_{4} (z))$ .

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Step 10.6.1

Combine $- 9$ and $\frac{1}{4}$ .

$\frac{\log_{4} (x)}{2} + \frac{5 \log_{4} (y)}{2} + \frac{- 9}{4} \log_{4} (z)$

Step 10.6.2

Combine $\frac{- 9}{4}$ and $\log_{4} (z)$ .

$\frac{\log_{4} (x)}{2} + \frac{5 \log_{4} (y)}{2} + \frac{- 9 \log_{4} (z)}{4}$

Step 11

Move the negative in front of the fraction.

$\frac{\log_{4} (x)}{2} + \frac{5 \log_{4} (y)}{2} - \frac{9 \log_{4} (z)}{4}$