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

$\log_{2} (\frac{\sqrt[3]{x}}{x^{2} + x})$

Step 1

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

$\log_{2} (\sqrt[3]{x}) - \log_{2} (x^{2} + x)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\log_{2} (x^{\frac{1}{3}}) - \log_{2} (x^{2} + x)$

Step 3

Expand $\log_{2} (x^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{2} (x) - \log_{2} (x^{2} + x)$

Step 4

Combine $\frac{1}{3}$ and $\log_{2} (x)$ .

$\frac{\log_{2} (x)}{3} - \log_{2} (x^{2} + x)$

Step 5

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

$\frac{\log_{2} (x)}{3} - \log_{2} (x^{2} + x) \cdot \frac{3}{3}$

Step 6

Combine $- \log_{2} (x^{2} + x)$ and $\frac{3}{3}$ .

$\frac{\log_{2} (x)}{3} + \frac{- \log_{2} (x^{2} + x) \cdot 3}{3}$

Step 7

Combine the numerators over the common denominator.

$\frac{\log_{2} (x) - \log_{2} (x^{2} + x) \cdot 3}{3}$

Step 8

Multiply $3$ by $- 1$ .

$\frac{\log_{2} (x) - 3 \log_{2} (x^{2} + x)}{3}$