Expand the Logarithmic Expression log of ( cube root of x^4)/((x+5)^2)

$\log (\frac{\sqrt[3]{x^{4}}}{{(x + 5)}^{2}})$

Rewrite $\log (\frac{\sqrt[3]{x^{4}}}{{(x + 5)}^{2}})$ as $\log (\sqrt[3]{x^{4}}) - \log ({(x + 5)}^{2})$ .

$\log (\sqrt[3]{x^{4}}) - \log ({(x + 5)}^{2})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{4}}$ as $x^{\frac{4}{3}}$ .

$\log (x^{\frac{4}{3}}) - \log ({(x + 5)}^{2})$

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

$\frac{4}{3} \log (x) - \log ({(x + 5)}^{2})$

Expand $\log ({(x + 5)}^{2})$ by moving $2$ outside the logarithm.

$\frac{4}{3} \log (x) - (2 \log (x + 5))$

Multiply $2$ by $- 1$ .

$\frac{4}{3} \log (x) - 2 \log (x + 5)$

Combine $\frac{4}{3}$ and $\log (x)$ .

$\frac{4 \log (x)}{3} - 2 \log (x + 5)$

To write $- 2 \log (x + 5)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 \log (x)}{3} - 2 \log (x + 5) \cdot \frac{3}{3}$

Combine $- 2 \log (x + 5)$ and $\frac{3}{3}$ .

$\frac{4 \log (x)}{3} + \frac{- 2 \log (x + 5) \cdot 3}{3}$

Combine the numerators over the common denominator.

$\frac{4 \log (x) - 2 \log (x + 5) \cdot 3}{3}$

Simplify the numerator.

Factor $2$ out of $4 \log (x) - 2 \log (x + 5) \cdot 3$ .

Factor $2$ out of $4 \log (x)$ .

$\frac{2 (2 \log (x)) - 2 \log (x + 5) \cdot 3}{3}$

Factor $2$ out of $- 2 \log (x + 5) \cdot 3$ .

$\frac{2 (2 \log (x)) + 2 (- \log (x + 5) \cdot 3)}{3}$

Factor $2$ out of $2 (2 \log (x)) + 2 (- \log (x + 5) \cdot 3)$ .

$\frac{2 (2 \log (x) - \log (x + 5) \cdot 3)}{3}$

Multiply $3$ by $- 1$ .

$\frac{2 (2 \log (x) - 3 \log (x + 5))}{3}$