Precalculus Examples

$\ln ({(\frac{x^{2} - x - 6}{{(x + 6)}^{4}})}^{\frac{1}{3}})$

Step 1

Expand $\ln ({(\frac{x^{2} - x - 6}{{(x + 6)}^{4}})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \ln (\frac{x^{2} - x - 6}{{(x + 6)}^{4}})$

Step 2

Rewrite $\ln (\frac{x^{2} - x - 6}{{(x + 6)}^{4}})$ as $\ln (x^{2} - x - 6) - \ln ({(x + 6)}^{4})$ .

$\frac{1}{3} (\ln (x^{2} - x - 6) - \ln ({(x + 6)}^{4}))$

Step 3

Expand $\ln ({(x + 6)}^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{3} (\ln (x^{2} - x - 6) - (4 \ln (x + 6)))$

Step 4

Multiply $4$ by $- 1$ .

$\frac{1}{3} (\ln (x^{2} - x - 6) - 4 \ln (x + 6))$

Step 5

Apply the distributive property.

$\frac{1}{3} \ln (x^{2} - x - 6) + \frac{1}{3} (- 4 \ln (x + 6))$

Step 6

Combine $\frac{1}{3}$ and $\ln (x^{2} - x - 6)$ .

$\frac{\ln (x^{2} - x - 6)}{3} + \frac{1}{3} (- 4 \ln (x + 6))$

Step 7

Multiply $\frac{1}{3} (- 4 \ln (x + 6))$ .

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Step 7.1

Combine $- 4$ and $\frac{1}{3}$ .

$\frac{\ln (x^{2} - x - 6)}{3} + \frac{- 4}{3} \ln (x + 6)$

Step 7.2

Combine $\frac{- 4}{3}$ and $\ln (x + 6)$ .

$\frac{\ln (x^{2} - x - 6)}{3} + \frac{- 4 \ln (x + 6)}{3}$

Step 8

Combine the numerators over the common denominator.

$\frac{\ln (x^{2} - x - 6) - 4 \ln (x + 6)}{3}$