Expand the Logarithmic Expression natural log of fifth root of x^5(x^4-4)

$\ln (\sqrt[5]{x^{5} (x^{4} - 4)})$

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\ln (x^{5} (x^{4} - 4))$ as $\ln (x^{5}) + \ln (x^{4} - 4)$ .

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

Step 4

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

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

Step 5

Apply the distributive property.

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

Step 6

Cancel the common factor of $5$ .

Factor $5$ out of $5 \ln (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

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

Step 8

To write $\ln (x)$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\ln (x) \cdot \frac{5}{5} + \frac{\ln (x^{4} - 4)}{5}$

Step 9

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

$\frac{\ln (x) \cdot 5}{5} + \frac{\ln (x^{4} - 4)}{5}$

Step 10

Combine the numerators over the common denominator.

$\frac{\ln (x) \cdot 5 + \ln (x^{4} - 4)}{5}$

Step 11

Move $5$ to the left of $\ln (x)$ .

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