Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4 x^{2 - 1}}$ as ${(4 x^{2 - 1})}^{\frac{1}{5}}$ .

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

Step 3

Subtract $1$ from $2$ .

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

Step 4

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

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

Step 5

Add $2$ and $1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Multiply $\ln (x)$ by $1$ .

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

Step 9

Rewrite $\ln (4 x^{3})$ as $\ln (4) + \ln (x^{3})$ .

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

Step 10

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

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

Step 11

Simplify each term.

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Step 11.1

Simplify each term.

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Step 11.1.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

Step 11.1.2

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

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Multiply $\frac{1}{5} (2 \ln (2))$ .

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Step 11.3.1

Combine $2$ and $\frac{1}{5}$ .

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

Step 11.3.2

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

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

Step 11.4

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

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

Step 11.5

Simplify each term.

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Step 11.5.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

Step 11.5.2

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

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

Step 11.6

Apply the distributive property.

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

Step 11.7

Multiply $2$ by $- 1$ .

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

Step 11.8

Multiply $3$ by $- 1$ .

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

Step 12

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

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

Step 13

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

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

Step 14

Combine the numerators over the common denominator.

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

Step 15

Simplify each term.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Multiply $5$ by $- 2$ .

$\frac{2 \ln (2) - 10 \ln (2)}{5} + \frac{\ln (x)}{5} - 3 \ln (x)$

Step 15.1.2

Subtract $10 \ln (2)$ from $2 \ln (2)$ .

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

Step 15.2

Move the negative in front of the fraction.

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

Step 16

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

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

Step 17

Combine $- 3 \ln (x)$ and $\frac{5}{5}$ .

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

Step 18

Combine the numerators over the common denominator.

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

Step 19

Combine the numerators over the common denominator.

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

Step 20

Simplify the numerator.

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Step 20.1

Multiply $5$ by $- 3$ .

$\frac{- 8 \ln (2) + \ln (x) - 15 \ln (x)}{5}$

Step 20.2

Subtract $15 \ln (x)$ from $\ln (x)$ .

$\frac{- 8 \ln (2) - 14 \ln (x)}{5}$

Step 21

Factor $- 2$ out of $- 8 \ln (2) - 14 \ln (x)$ .

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Step 21.1

Factor $- 2$ out of $- 8 \ln (2)$ .

$\frac{- 2 (4 \ln (2)) - 14 \ln (x)}{5}$

Step 21.2

Factor $- 2$ out of $- 14 \ln (x)$ .

$\frac{- 2 (4 \ln (2)) - 2 (7 \ln (x))}{5}$

Step 21.3

Factor $- 2$ out of $- 2 (4 \ln (2)) - 2 (7 \ln (x))$ .

$\frac{- 2 (4 \ln (2) + 7 \ln (x))}{5}$

Step 22

Move the negative in front of the fraction.

$- \frac{2 (4 \ln (2) + 7 \ln (x))}{5}$