Precalculus Examples

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

Step 1

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

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

Step 2

Add $2$ and $1$ .

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

Step 3

Expand $\ln ({(\frac{4 x^{2} - 1}{4 x^{3}})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

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

Step 4

Rewrite $\ln (\frac{4 x^{2} - 1}{4 x^{3}})$ as $\ln (4 x^{2} - 1) - \ln (4 x^{3})$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Multiply $2$ by $- 1$ .

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

Step 7.4

Multiply $3$ by $- 1$ .

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{5}$ and $\ln (4 x^{2} - 1)$ .

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

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.3

Multiply $\frac{1}{5} (- 3 \ln (x))$ .

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Step 9.3.1

Combine $- 3$ and $\frac{1}{5}$ .

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

Step 9.3.2

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

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

Step 10

Simplify each term.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

Move the negative in front of the fraction.

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

Step 11

Combine the numerators over the common denominator.

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