Trigonometry Examples

$\log_{c} (\sqrt[4]{\frac{x^{7}}{y^{4} z^{5}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{x^{7}}{y^{4} z^{5}}}$ as ${(\frac{x^{7}}{y^{4} z^{5}})}^{\frac{1}{4}}$ .

$\log_{c} ({(\frac{x^{7}}{y^{4} z^{5}})}^{\frac{1}{4}})$

Step 2

Expand $\log_{c} ({(\frac{x^{7}}{y^{4} z^{5}})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{c} (\frac{x^{7}}{y^{4} z^{5}})$

Step 3

Rewrite $\log_{c} (\frac{x^{7}}{y^{4} z^{5}})$ as $\log_{c} (x^{7}) - \log_{c} (y^{4} z^{5})$ .

$\frac{1}{4} (\log_{c} (x^{7}) - \log_{c} (y^{4} z^{5}))$

Step 4

Expand $\log_{c} (x^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{4} (7 \log_{c} (x) - \log_{c} (y^{4} z^{5}))$

Step 5

Rewrite $\log_{c} (y^{4} z^{5})$ as $\log_{c} (y^{4}) + \log_{c} (z^{5})$ .

$\frac{1}{4} (7 \log_{c} (x) - (\log_{c} (y^{4}) + \log_{c} (z^{5})))$

Step 6

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

$\frac{1}{4} (7 \log_{c} (x) - (4 \log_{c} (y) + \log_{c} (z^{5})))$

Step 7

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

$\frac{1}{4} (7 \log_{c} (x) - (4 \log_{c} (y) + 5 \log_{c} (z)))$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{4} (7 \log_{c} (x) - (4 \log_{c} (y)) - (5 \log_{c} (z)))$

Step 8.2

Multiply $4$ by $- 1$ .

$\frac{1}{4} (7 \log_{c} (x) - 4 \log_{c} (y) - (5 \log_{c} (z)))$

Step 8.3

Multiply $5$ by $- 1$ .

$\frac{1}{4} (7 \log_{c} (x) - 4 \log_{c} (y) - 5 \log_{c} (z))$

Step 9

Apply the distributive property.

$\frac{1}{4} (7 \log_{c} (x)) + \frac{1}{4} (- 4 \log_{c} (y)) + \frac{1}{4} (- 5 \log_{c} (z))$

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{4} (7 \log_{c} (x))$ .

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Step 10.1.1

Combine $7$ and $\frac{1}{4}$ .

$\frac{7}{4} \log_{c} (x) + \frac{1}{4} (- 4 \log_{c} (y)) + \frac{1}{4} (- 5 \log_{c} (z))$

Step 10.1.2

Combine $\frac{7}{4}$ and $\log_{c} (x)$ .

$\frac{7 \log_{c} (x)}{4} + \frac{1}{4} (- 4 \log_{c} (y)) + \frac{1}{4} (- 5 \log_{c} (z))$

Step 10.2

Cancel the common factor of $4$ .

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Step 10.2.1

Factor $4$ out of $- 4 \log_{c} (y)$ .

$\frac{7 \log_{c} (x)}{4} + \frac{1}{4} (4 (- \log_{c} (y))) + \frac{1}{4} (- 5 \log_{c} (z))$

Step 10.2.2

Cancel the common factor.

$\frac{7 \log_{c} (x)}{4} + \frac{1}{4} (4 (- \log_{c} (y))) + \frac{1}{4} (- 5 \log_{c} (z))$

Step 10.2.3

Rewrite the expression.

$\frac{7 \log_{c} (x)}{4} - \log_{c} (y) + \frac{1}{4} (- 5 \log_{c} (z))$

Step 10.3

Multiply $\frac{1}{4} (- 5 \log_{c} (z))$ .

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Step 10.3.1

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

$\frac{7 \log_{c} (x)}{4} - \log_{c} (y) + \frac{- 5}{4} \log_{c} (z)$

Step 10.3.2

Combine $\frac{- 5}{4}$ and $\log_{c} (z)$ .

$\frac{7 \log_{c} (x)}{4} - \log_{c} (y) + \frac{- 5 \log_{c} (z)}{4}$

Step 11

Move the negative in front of the fraction.

$\frac{7 \log_{c} (x)}{4} - \log_{c} (y) - \frac{5 \log_{c} (z)}{4}$