Precalculus Examples

$\ln (\frac{x^{3} \sqrt{x + 8}}{\sqrt[7]{x^{2} + 2 x + 1}})$

Step 1

Rewrite $\ln (\frac{x^{3} \sqrt{x + 8}}{\sqrt[7]{x^{2} + 2 x + 1}})$ as $\ln (x^{3} \sqrt{x + 8}) - \ln (\sqrt[7]{x^{2} + 2 x + 1})$ .

$\ln (x^{3} \sqrt{x + 8}) - \ln (\sqrt[7]{x^{2} + 2 x + 1})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 8}$ as ${(x + 8)}^{\frac{1}{2}}$ .

$\ln (x^{3} {(x + 8)}^{\frac{1}{2}}) - \ln (\sqrt[7]{x^{2} + 2 x + 1})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{2} + 2 x + 1}$ as ${(x^{2} + 2 x + 1)}^{\frac{1}{7}}$ .

$\ln (x^{3} {(x + 8)}^{\frac{1}{2}}) - \ln ({(x^{2} + 2 x + 1)}^{\frac{1}{7}})$

Step 4

Expand $\ln ({(x^{2} + 2 x + 1)}^{\frac{1}{7}})$ by moving $\frac{1}{7}$ outside the logarithm.

$\ln (x^{3} {(x + 8)}^{\frac{1}{2}}) - (\frac{1}{7} \ln (x^{2} + 2 x + 1))$

Step 5

Combine $\frac{1}{7}$ and $\ln (x^{2} + 2 x + 1)$ .

$\ln (x^{3} {(x + 8)}^{\frac{1}{2}}) - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 6

Rewrite $\ln (x^{3} {(x + 8)}^{\frac{1}{2}})$ as $\ln (x^{3}) + \ln ({(x + 8)}^{\frac{1}{2}})$ .

$\ln (x^{3}) + \ln ({(x + 8)}^{\frac{1}{2}}) - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 7

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

$3 \ln (x) + \ln ({(x + 8)}^{\frac{1}{2}}) - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 8

Expand $\ln ({(x + 8)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 \ln (x) + \frac{1}{2} \ln (x + 8) - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 9

Combine $\frac{1}{2}$ and $\ln (x + 8)$ .

$3 \ln (x) + \frac{\ln (x + 8)}{2} - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 10

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

$3 \ln (x) \cdot \frac{2}{2} + \frac{\ln (x + 8)}{2} - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 11

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

$\frac{3 \ln (x) \cdot 2}{2} + \frac{\ln (x + 8)}{2} - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 12

Combine the numerators over the common denominator.

$\frac{3 \ln (x) \cdot 2 + \ln (x + 8)}{2} - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 13

Multiply $2$ by $3$ .

$\frac{6 \ln (x) + \ln (x + 8)}{2} - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 14

To write $\frac{6 \ln (x) + \ln (x + 8)}{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{6 \ln (x) + \ln (x + 8)}{2} \cdot \frac{7}{7} - \frac{\ln (x^{2} + 2 x + 1)}{7}$

Step 15

To write $- \frac{\ln (x^{2} + 2 x + 1)}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{6 \ln (x) + \ln (x + 8)}{2} \cdot \frac{7}{7} - \frac{\ln (x^{2} + 2 x + 1)}{7} \cdot \frac{2}{2}$

Step 16

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.1

Multiply $\frac{6 \ln (x) + \ln (x + 8)}{2}$ by $\frac{7}{7}$ .

$\frac{(6 \ln (x) + \ln (x + 8)) \cdot 7}{2 \cdot 7} - \frac{\ln (x^{2} + 2 x + 1)}{7} \cdot \frac{2}{2}$

Step 16.2

Multiply $2$ by $7$ .

$\frac{(6 \ln (x) + \ln (x + 8)) \cdot 7}{14} - \frac{\ln (x^{2} + 2 x + 1)}{7} \cdot \frac{2}{2}$

Step 16.3

Multiply $\frac{\ln (x^{2} + 2 x + 1)}{7}$ by $\frac{2}{2}$ .

$\frac{(6 \ln (x) + \ln (x + 8)) \cdot 7}{14} - \frac{\ln (x^{2} + 2 x + 1) \cdot 2}{7 \cdot 2}$

Step 16.4

Multiply $7$ by $2$ .

$\frac{(6 \ln (x) + \ln (x + 8)) \cdot 7}{14} - \frac{\ln (x^{2} + 2 x + 1) \cdot 2}{14}$

Step 17

Combine the numerators over the common denominator.

$\frac{(6 \ln (x) + \ln (x + 8)) \cdot 7 - \ln (x^{2} + 2 x + 1) \cdot 2}{14}$

Step 18

Simplify the numerator.

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Step 18.1

Apply the distributive property.

$\frac{6 \ln (x) \cdot 7 + \ln (x + 8) \cdot 7 - \ln (x^{2} + 2 x + 1) \cdot 2}{14}$

Step 18.2

Multiply $7$ by $6$ .

$\frac{42 \ln (x) + \ln (x + 8) \cdot 7 - \ln (x^{2} + 2 x + 1) \cdot 2}{14}$

Step 18.3

Move $7$ to the left of $\ln (x + 8)$ .

$\frac{42 \ln (x) + 7 \cdot \ln (x + 8) - \ln (x^{2} + 2 x + 1) \cdot 2}{14}$

Step 18.4

Multiply $2$ by $- 1$ .

$\frac{42 \ln (x) + 7 \ln (x + 8) - 2 \ln (x^{2} + 2 x + 1)}{14}$