Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $2$ by $- 1$ .

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

Step 5

Apply the distributive property.

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

Step 6

Combine $\frac{1}{3}$ and $\ln (x^{2} - x - 2)$ .

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

Step 7

Multiply $\frac{1}{3} (- 2 \ln (x + 3))$ .

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Step 7.1

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 7.2

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

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

Step 8

Combine the numerators over the common denominator.

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