Precalculus Examples

$\ln (\frac{x^{4} \sqrt{x^{2} - 4}}{\sqrt[3]{x^{4} + 5}})$

Step 1

Rewrite $\ln (\frac{x^{4} \sqrt{x^{2} - 4}}{\sqrt[3]{x^{4} + 5}})$ as $\ln (x^{4} \sqrt{x^{2} - 4}) - \ln (\sqrt[3]{x^{4} + 5})$ .

$\ln (x^{4} \sqrt{x^{2} - 4}) - \ln (\sqrt[3]{x^{4} + 5})$

Step 2

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

$\ln (x^{4} {(x^{2} - 4)}^{\frac{1}{2}}) - \ln (\sqrt[3]{x^{4} + 5})$

Step 3

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

$\ln (x^{4} {(x^{2} - 4)}^{\frac{1}{2}}) - \ln ({(x^{4} + 5)}^{\frac{1}{3}})$

Step 4

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

$\ln (x^{4} {(x^{2} - 4)}^{\frac{1}{2}}) - (\frac{1}{3} \ln (x^{4} + 5))$

Step 5

Combine $\frac{1}{3}$ and $\ln (x^{4} + 5)$ .

$\ln (x^{4} {(x^{2} - 4)}^{\frac{1}{2}}) - \frac{\ln (x^{4} + 5)}{3}$

Step 6

Rewrite $\ln (x^{4} {(x^{2} - 4)}^{\frac{1}{2}})$ as $\ln (x^{4}) + \ln ({(x^{2} - 4)}^{\frac{1}{2}})$ .

$\ln (x^{4}) + \ln ({(x^{2} - 4)}^{\frac{1}{2}}) - \frac{\ln (x^{4} + 5)}{3}$

Step 7

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

$4 \ln (x) + \ln ({(x^{2} - 4)}^{\frac{1}{2}}) - \frac{\ln (x^{4} + 5)}{3}$

Step 8

Expand $\ln ({(x^{2} - 4)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$4 \ln (x) + \frac{1}{2} \ln (x^{2} - 4) - \frac{\ln (x^{4} + 5)}{3}$

Step 9

Combine $\frac{1}{2}$ and $\ln (x^{2} - 4)$ .

$4 \ln (x) + \frac{\ln (x^{2} - 4)}{2} - \frac{\ln (x^{4} + 5)}{3}$

Step 10

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

$4 \ln (x) \cdot \frac{2}{2} + \frac{\ln (x^{2} - 4)}{2} - \frac{\ln (x^{4} + 5)}{3}$

Step 11

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

$\frac{4 \ln (x) \cdot 2}{2} + \frac{\ln (x^{2} - 4)}{2} - \frac{\ln (x^{4} + 5)}{3}$

Step 12

Combine the numerators over the common denominator.

$\frac{4 \ln (x) \cdot 2 + \ln (x^{2} - 4)}{2} - \frac{\ln (x^{4} + 5)}{3}$

Step 13

Multiply $2$ by $4$ .

$\frac{8 \ln (x) + \ln (x^{2} - 4)}{2} - \frac{\ln (x^{4} + 5)}{3}$

Step 14

To write $\frac{8 \ln (x) + \ln (x^{2} - 4)}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8 \ln (x) + \ln (x^{2} - 4)}{2} \cdot \frac{3}{3} - \frac{\ln (x^{4} + 5)}{3}$

Step 15

To write $- \frac{\ln (x^{4} + 5)}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{8 \ln (x) + \ln (x^{2} - 4)}{2} \cdot \frac{3}{3} - \frac{\ln (x^{4} + 5)}{3} \cdot \frac{2}{2}$

Step 16

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

Tap for more steps...

Step 16.1

Multiply $\frac{8 \ln (x) + \ln (x^{2} - 4)}{2}$ by $\frac{3}{3}$ .

$\frac{(8 \ln (x) + \ln (x^{2} - 4)) \cdot 3}{2 \cdot 3} - \frac{\ln (x^{4} + 5)}{3} \cdot \frac{2}{2}$

Step 16.2

Multiply $2$ by $3$ .

$\frac{(8 \ln (x) + \ln (x^{2} - 4)) \cdot 3}{6} - \frac{\ln (x^{4} + 5)}{3} \cdot \frac{2}{2}$

Step 16.3

Multiply $\frac{\ln (x^{4} + 5)}{3}$ by $\frac{2}{2}$ .

$\frac{(8 \ln (x) + \ln (x^{2} - 4)) \cdot 3}{6} - \frac{\ln (x^{4} + 5) \cdot 2}{3 \cdot 2}$

Step 16.4

Multiply $3$ by $2$ .

$\frac{(8 \ln (x) + \ln (x^{2} - 4)) \cdot 3}{6} - \frac{\ln (x^{4} + 5) \cdot 2}{6}$

Step 17

Combine the numerators over the common denominator.

$\frac{(8 \ln (x) + \ln (x^{2} - 4)) \cdot 3 - \ln (x^{4} + 5) \cdot 2}{6}$

Step 18

Simplify the numerator.

Tap for more steps...

Step 18.1

Apply the distributive property.

$\frac{8 \ln (x) \cdot 3 + \ln (x^{2} - 4) \cdot 3 - \ln (x^{4} + 5) \cdot 2}{6}$

Step 18.2

Multiply $3$ by $8$ .

$\frac{24 \ln (x) + \ln (x^{2} - 4) \cdot 3 - \ln (x^{4} + 5) \cdot 2}{6}$

Step 18.3

Move $3$ to the left of $\ln (x^{2} - 4)$ .

$\frac{24 \ln (x) + 3 \cdot \ln (x^{2} - 4) - \ln (x^{4} + 5) \cdot 2}{6}$

Step 18.4

Multiply $2$ by $- 1$ .

$\frac{24 \ln (x) + 3 \ln (x^{2} - 4) - 2 \ln (x^{4} + 5)}{6}$