Precalculus Examples

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

Step 1

Rewrite $\ln (\frac{\sqrt{x + 5} \sqrt[5]{x - 2}}{{(x + 1)}^{2}})$ as $\ln (\sqrt{x + 5} \sqrt[5]{x - 2}) - \ln ({(x + 1)}^{2})$ .

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 5}$ as ${(x + 5)}^{\frac{1}{2}}$ .

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

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x - 2}$ as ${(x - 2)}^{\frac{1}{5}}$ .

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

Step 4

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

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

Rewrite $\ln ({(x + 5)}^{\frac{1}{2}} {(x - 2)}^{\frac{1}{5}})$ as $\ln ({(x + 5)}^{\frac{1}{2}}) + \ln ({(x - 2)}^{\frac{1}{5}})$ .

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

Step 7

Expand $\ln ({(x + 5)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 8

Expand $\ln ({(x - 2)}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

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

Step 9

Simplify each term.

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Step 9.1

Combine $\frac{1}{2}$ and $\ln (x + 5)$ .

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

Step 9.2

Combine $\frac{1}{5}$ and $\ln (x - 2)$ .

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

Step 10

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

$\frac{\ln (x + 5)}{2} \cdot \frac{5}{5} + \frac{\ln (x - 2)}{5} - 2 \ln (x + 1)$

Step 11

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

$\frac{\ln (x + 5)}{2} \cdot \frac{5}{5} + \frac{\ln (x - 2)}{5} \cdot \frac{2}{2} - 2 \ln (x + 1)$

Step 12

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.1

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

$\frac{\ln (x + 5) \cdot 5}{2 \cdot 5} + \frac{\ln (x - 2)}{5} \cdot \frac{2}{2} - 2 \ln (x + 1)$

Step 12.2

Multiply $2$ by $5$ .

$\frac{\ln (x + 5) \cdot 5}{10} + \frac{\ln (x - 2)}{5} \cdot \frac{2}{2} - 2 \ln (x + 1)$

Step 12.3

Multiply $\frac{\ln (x - 2)}{5}$ by $\frac{2}{2}$ .

$\frac{\ln (x + 5) \cdot 5}{10} + \frac{\ln (x - 2) \cdot 2}{5 \cdot 2} - 2 \ln (x + 1)$

Step 12.4

Multiply $5$ by $2$ .

$\frac{\ln (x + 5) \cdot 5}{10} + \frac{\ln (x - 2) \cdot 2}{10} - 2 \ln (x + 1)$

Step 13

Combine the numerators over the common denominator.

$\frac{\ln (x + 5) \cdot 5 + \ln (x - 2) \cdot 2}{10} - 2 \ln (x + 1)$

Step 14

Simplify the numerator.

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Step 14.1

Move $5$ to the left of $\ln (x + 5)$ .

$\frac{5 \cdot \ln (x + 5) + \ln (x - 2) \cdot 2}{10} - 2 \ln (x + 1)$

Step 14.2

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

$\frac{5 \ln (x + 5) + 2 \ln (x - 2)}{10} - 2 \ln (x + 1)$

Step 15

To write $- 2 \ln (x + 1)$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{5 \ln (x + 5) + 2 \ln (x - 2)}{10} - 2 \ln (x + 1) \cdot \frac{10}{10}$

Step 16

Combine $- 2 \ln (x + 1)$ and $\frac{10}{10}$ .

$\frac{5 \ln (x + 5) + 2 \ln (x - 2)}{10} + \frac{- 2 \ln (x + 1) \cdot 10}{10}$

Step 17

Combine the numerators over the common denominator.

$\frac{5 \ln (x + 5) + 2 \ln (x - 2) - 2 \ln (x + 1) \cdot 10}{10}$

Step 18

Multiply $10$ by $- 2$ .

$\frac{5 \ln (x + 5) + 2 \ln (x - 2) - 20 \ln (x + 1)}{10}$