Precalculus Examples

$\ln (\frac{\sqrt[4]{x^{3} w^{5}}}{y^{9}})$

Step 1

Rewrite $\ln (\frac{\sqrt[4]{x^{3} w^{5}}}{y^{9}})$ as $\ln (\sqrt[4]{x^{3} w^{5}}) - \ln (y^{9})$ .

$\ln (\sqrt[4]{x^{3} w^{5}}) - \ln (y^{9})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3} w^{5}}$ as ${(x^{3} w^{5})}^{\frac{1}{4}}$ .

$\ln ({(x^{3} w^{5})}^{\frac{1}{4}}) - \ln (y^{9})$

Step 3

Expand $\ln ({(x^{3} w^{5})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \ln (x^{3} w^{5}) - \ln (y^{9})$

Step 4

Expand $\ln (y^{9})$ by moving $9$ outside the logarithm.

$\frac{1}{4} \ln (x^{3} w^{5}) - (9 \ln (y))$

Step 5

Multiply $9$ by $- 1$ .

$\frac{1}{4} \ln (x^{3} w^{5}) - 9 \ln (y)$

Step 6

Rewrite $\ln (x^{3} w^{5})$ as $\ln (x^{3}) + \ln (w^{5})$ .

$\frac{1}{4} (\ln (x^{3}) + \ln (w^{5})) - 9 \ln (y)$

Step 7

Expand $\ln (x^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{4} (3 \ln (x) + \ln (w^{5})) - 9 \ln (y)$

Step 8

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

$\frac{1}{4} (3 \ln (x) + 5 \ln (w)) - 9 \ln (y)$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$\frac{1}{4} (3 \ln (x)) + \frac{1}{4} (5 \ln (w)) - 9 \ln (y)$

Step 9.2

Multiply $\frac{1}{4} (3 \ln (x))$ .

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Step 9.2.1

Combine $3$ and $\frac{1}{4}$ .

$\frac{3}{4} \ln (x) + \frac{1}{4} (5 \ln (w)) - 9 \ln (y)$

Step 9.2.2

Combine $\frac{3}{4}$ and $\ln (x)$ .

$\frac{3 \ln (x)}{4} + \frac{1}{4} (5 \ln (w)) - 9 \ln (y)$

Step 9.3

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

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Step 9.3.1

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

$\frac{3 \ln (x)}{4} + \frac{5}{4} \ln (w) - 9 \ln (y)$

Step 9.3.2

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

$\frac{3 \ln (x)}{4} + \frac{5 \ln (w)}{4} - 9 \ln (y)$