Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $5$ by $- 1$ .

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.2

Multiply $\frac{1}{4} (- 5 \ln (y))$ .

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Step 9.2.1

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

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

Step 9.2.2

Combine $\frac{- 5}{4}$ and $\ln (y)$ .

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

Step 9.3

Combine $\frac{1}{4}$ and $\ln (z)$ .

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

Step 10

Move the negative in front of the fraction.

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