Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $3$ by $- 1$ .

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

Step 5

Apply the distributive property.

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

Step 6

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

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

Step 7

Cancel the common factor of $3$ .

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Step 7.1

Factor $3$ out of $- 3 \ln (x^{2} - 4)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Multiply $- 1$ by $2$ .

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

Step 9

To write $- 2 \ln (x^{2} - 4)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10

Combine $- 2 \ln (x^{2} - 4)$ and $\frac{3}{3}$ .

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Simplify the numerator.

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Step 12.1

Factor $2$ out of $2 \ln (x - 2) - 2 \ln (x^{2} - 4) \cdot 3$ .

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Step 12.1.1

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

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

Step 12.1.2

Factor $2$ out of $- 2 \ln (x^{2} - 4) \cdot 3$ .

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

Step 12.1.3

Factor $2$ out of $2 (\ln (x - 2)) + 2 (- \ln (x^{2} - 4) \cdot 3)$ .

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

Step 12.2

Multiply $3$ by $- 1$ .

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