Precalculus Examples

$\ln (\frac{\sqrt[3]{x^{4} - 81}}{(x + 16) \sqrt{x - 3}})$

Step 1

Rewrite $\ln (\frac{\sqrt[3]{x^{4} - 81}}{(x + 16) \sqrt{x - 3}})$ as $\ln (\sqrt[3]{x^{4} - 81}) - \ln ((x + 16) \sqrt{x - 3})$ .

$\ln (\sqrt[3]{x^{4} - 81}) - \ln ((x + 16) \sqrt{x - 3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{4} - 81}$ as ${(x^{4} - 81)}^{\frac{1}{3}}$ .

$\ln ({(x^{4} - 81)}^{\frac{1}{3}}) - \ln ((x + 16) \sqrt{x - 3})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

$\ln ({(x^{4} - 81)}^{\frac{1}{3}}) - \ln ((x + 16) {(x - 3)}^{\frac{1}{2}})$

Step 4

Expand $\ln ({(x^{4} - 81)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \ln (x^{4} - 81) - \ln ((x + 16) {(x - 3)}^{\frac{1}{2}})$

Step 5

Rewrite $\ln ((x + 16) {(x - 3)}^{\frac{1}{2}})$ as $\ln (x + 16) + \ln ({(x - 3)}^{\frac{1}{2}})$ .

$\frac{1}{3} \ln (x^{4} - 81) - (\ln (x + 16) + \ln ({(x - 3)}^{\frac{1}{2}}))$

Step 6

Expand $\ln ({(x - 3)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{3} \ln (x^{4} - 81) - (\ln (x + 16) + \frac{1}{2} \ln (x - 3))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{1}{3}$ and $\ln (x^{4} - 81)$ .

$\frac{\ln (x^{4} - 81)}{3} - (\ln (x + 16) + \frac{1}{2} \ln (x - 3))$

Step 7.2

Combine $\frac{1}{2}$ and $\ln (x - 3)$ .

$\frac{\ln (x^{4} - 81)}{3} - (\ln (x + 16) + \frac{\ln (x - 3)}{2})$

Step 7.3

To write $\ln (x + 16)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\ln (x^{4} - 81)}{3} - (\ln (x + 16) \cdot \frac{2}{2} + \frac{\ln (x - 3)}{2})$

Step 7.4

Combine $\ln (x + 16)$ and $\frac{2}{2}$ .

$\frac{\ln (x^{4} - 81)}{3} - (\frac{\ln (x + 16) \cdot 2}{2} + \frac{\ln (x - 3)}{2})$

Step 7.5

Combine the numerators over the common denominator.

$\frac{\ln (x^{4} - 81)}{3} - \frac{\ln (x + 16) \cdot 2 + \ln (x - 3)}{2}$

Step 7.6

Move $2$ to the left of $\ln (x + 16)$ .

$\frac{\ln (x^{4} - 81)}{3} - \frac{2 \ln (x + 16) + \ln (x - 3)}{2}$

Step 8

To write $\frac{\ln (x^{4} - 81)}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\ln (x^{4} - 81)}{3} \cdot \frac{2}{2} - \frac{2 \ln (x + 16) + \ln (x - 3)}{2}$

Step 9

To write $- \frac{2 \ln (x + 16) + \ln (x - 3)}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\ln (x^{4} - 81)}{3} \cdot \frac{2}{2} - \frac{2 \ln (x + 16) + \ln (x - 3)}{2} \cdot \frac{3}{3}$

Step 10

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{\ln (x^{4} - 81)}{3}$ by $\frac{2}{2}$ .

$\frac{\ln (x^{4} - 81) \cdot 2}{3 \cdot 2} - \frac{2 \ln (x + 16) + \ln (x - 3)}{2} \cdot \frac{3}{3}$

Step 10.2

Multiply $3$ by $2$ .

$\frac{\ln (x^{4} - 81) \cdot 2}{6} - \frac{2 \ln (x + 16) + \ln (x - 3)}{2} \cdot \frac{3}{3}$

Step 10.3

Multiply $\frac{2 \ln (x + 16) + \ln (x - 3)}{2}$ by $\frac{3}{3}$ .

$\frac{\ln (x^{4} - 81) \cdot 2}{6} - \frac{(2 \ln (x + 16) + \ln (x - 3)) \cdot 3}{2 \cdot 3}$

Step 10.4

Multiply $2$ by $3$ .

$\frac{\ln (x^{4} - 81) \cdot 2}{6} - \frac{(2 \ln (x + 16) + \ln (x - 3)) \cdot 3}{6}$

Step 11

Combine the numerators over the common denominator.

$\frac{\ln (x^{4} - 81) \cdot 2 - (2 \ln (x + 16) + \ln (x - 3)) \cdot 3}{6}$

Step 12

Simplify the numerator.

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Step 12.1

Move $2$ to the left of $\ln (x^{4} - 81)$ .

$\frac{2 \cdot \ln (x^{4} - 81) - (2 \ln (x + 16) + \ln (x - 3)) \cdot 3}{6}$

Step 12.2

Apply the distributive property.

$\frac{2 \ln (x^{4} - 81) + (- (2 \ln (x + 16)) - \ln (x - 3)) \cdot 3}{6}$

Step 12.3

Multiply $2$ by $- 1$ .

$\frac{2 \ln (x^{4} - 81) + (- 2 \ln (x + 16) - \ln (x - 3)) \cdot 3}{6}$

Step 12.4

Apply the distributive property.

$\frac{2 \ln (x^{4} - 81) - 2 \ln (x + 16) \cdot 3 - \ln (x - 3) \cdot 3}{6}$

Step 12.5

Multiply $3$ by $- 2$ .

$\frac{2 \ln (x^{4} - 81) - 6 \ln (x + 16) - \ln (x - 3) \cdot 3}{6}$

Step 12.6

Multiply $3$ by $- 1$ .

$\frac{2 \ln (x^{4} - 81) - 6 \ln (x + 16) - 3 \ln (x - 3)}{6}$