Precalculus Examples

$\ln ({(\frac{3 + x}{x^{5}})}^{\frac{1}{6}})$

Step 1

Expand $\ln ({(\frac{3 + x}{x^{5}})}^{\frac{1}{6}})$ by moving $\frac{1}{6}$ outside the logarithm.

$\frac{1}{6} \ln (\frac{3 + x}{x^{5}})$

Step 2

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

$\frac{1}{6} (\ln (3 + x) - \ln (x^{5}))$

Step 3

Expand $\ln (x^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{6} (\ln (3 + x) - (5 \ln (x)))$

Step 4

Multiply $5$ by $- 1$ .

$\frac{1}{6} (\ln (3 + x) - 5 \ln (x))$

Step 5

Apply the distributive property.

$\frac{1}{6} \ln (3 + x) + \frac{1}{6} (- 5 \ln (x))$

Step 6

Combine $\frac{1}{6}$ and $\ln (3 + x)$ .

$\frac{\ln (3 + x)}{6} + \frac{1}{6} (- 5 \ln (x))$

Step 7

Multiply $\frac{1}{6} (- 5 \ln (x))$ .

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Step 7.1

Combine $- 5$ and $\frac{1}{6}$ .

$\frac{\ln (3 + x)}{6} + \frac{- 5}{6} \ln (x)$

Step 7.2

Combine $\frac{- 5}{6}$ and $\ln (x)$ .

$\frac{\ln (3 + x)}{6} + \frac{- 5 \ln (x)}{6}$

Step 8

Combine the numerators over the common denominator.

$\frac{\ln (3 + x) - 5 \ln (x)}{6}$