Expand the Logarithmic Expression log of ( cube root of x^2)/((x+8)^5)

$\log (\frac{\sqrt[3]{x^{2}}}{{(x + 8)}^{5}})$

Rewrite $\log (\frac{\sqrt[3]{x^{2}}}{{(x + 8)}^{5}})$ as $\log (\sqrt[3]{x^{2}}) - \log ({(x + 8)}^{5})$ .

$\log (\sqrt[3]{x^{2}}) - \log ({(x + 8)}^{5})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\log (x^{\frac{2}{3}}) - \log ({(x + 8)}^{5})$

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

$\frac{2}{3} \log (x) - \log ({(x + 8)}^{5})$

Expand $\log ({(x + 8)}^{5})$ by moving $5$ outside the logarithm.

$\frac{2}{3} \log (x) - (5 \log (x + 8))$

Multiply $5$ by $- 1$ .

$\frac{2}{3} \log (x) - 5 \log (x + 8)$

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

$\frac{2 \log (x)}{3} - 5 \log (x + 8)$

To write $- 5 \log (x + 8)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 \log (x)}{3} - 5 \log (x + 8) \cdot \frac{3}{3}$

Combine $- 5 \log (x + 8)$ and $\frac{3}{3}$ .

$\frac{2 \log (x)}{3} + \frac{- 5 \log (x + 8) \cdot 3}{3}$

Combine the numerators over the common denominator.

$\frac{2 \log (x) - 5 \log (x + 8) \cdot 3}{3}$

Multiply $3$ by $- 5$ .

$\frac{2 \log (x) - 15 \log (x + 8)}{3}$