Precalculus Examples

$\ln (\sqrt[4]{\frac{3 (2 x + 1)}{(2 (x - 5)) (8 (x^{2} + 1))}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{3 (2 x + 1)}{(2 (x - 5)) (8 (x^{2} + 1))}}$ as ${(\frac{3 (2 x + 1)}{(2 (x - 5)) (8 (x^{2} + 1))})}^{\frac{1}{4}}$ .

$\ln ({(\frac{3 (2 x + 1)}{(2 (x - 5)) (8 (x^{2} + 1))})}^{\frac{1}{4}})$

Step 2

Expand $\ln ({(\frac{3 (2 x + 1)}{(2 (x - 5)) (8 (x^{2} + 1))})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \ln (\frac{3 (2 x + 1)}{(2 (x - 5)) (8 (x^{2} + 1))})$

Step 3

Multiply $8$ by $2$ .

$\frac{1}{4} \ln (\frac{3 (2 x + 1)}{16 (x - 5) (x^{2} + 1)})$

Step 4

Rewrite $\ln (\frac{3 (2 x + 1)}{16 (x - 5) (x^{2} + 1)})$ as $\ln (3 (2 x + 1)) - \ln (16 (x - 5) (x^{2} + 1))$ .

$\frac{1}{4} (\ln (3 (2 x + 1)) - \ln (16 (x - 5) (x^{2} + 1)))$

Step 5

Rewrite $\ln (3 (2 x + 1))$ as $\ln (3) + \ln (2 x + 1)$ .

$\frac{1}{4} (\ln (3) + \ln (2 x + 1) - \ln (16 (x - 5) (x^{2} + 1)))$

Step 6

Rewrite $\ln (16 (x - 5) (x^{2} + 1))$ as $\ln (16 (x - 5)) + \ln (x^{2} + 1)$ .

$\frac{1}{4} (\ln (3) + \ln (2 x + 1) - (\ln (16 (x - 5)) + \ln (x^{2} + 1)))$

Step 7

Rewrite $\ln (16 (x - 5))$ as $\ln (16) + \ln (x - 5)$ .

$\frac{1}{4} (\ln (3) + \ln (2 x + 1) - (\ln (16) + \ln (x - 5) + \ln (x^{2} + 1)))$

Step 8

Simplify each term.

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite $\ln (16)$ as $\ln (2^{4})$ .

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

Step 8.1.2

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

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Multiply $4$ by $- 1$ .

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

Step 9

Apply the distributive property.

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

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{4}$ and $\ln (3)$ .

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

Step 10.2

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

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

Step 10.3

Cancel the common factor of $4$ .

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Step 10.3.1

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

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

Step 10.3.2

Cancel the common factor.

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

Step 10.3.3

Rewrite the expression.

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

Step 10.4

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

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

Step 10.5

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

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Combine the numerators over the common denominator.

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