Expand the Logarithmic Expression log base 2 of p cube root of xy

$\log_{2} (p \sqrt[3]{x y})$

Step 1

Rewrite $\log_{2} (p \sqrt[3]{x y})$ as $\log_{2} (p) + \log_{2} (\sqrt[3]{x y})$ .

$\log_{2} (p) + \log_{2} (\sqrt[3]{x y})$

Step 2

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

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

Step 3

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

$\log_{2} (p) + \frac{1}{3} \log_{2} (x y)$

Step 4

Rewrite $\log_{2} (x y)$ as $\log_{2} (x) + \log_{2} (y)$ .

$\log_{2} (p) + \frac{1}{3} (\log_{2} (x) + \log_{2} (y))$

Step 5

Simplify each term.

Apply the distributive property.

$\log_{2} (p) + \frac{1}{3} \log_{2} (x) + \frac{1}{3} \log_{2} (y)$

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

$\log_{2} (p) + \frac{\log_{2} (x)}{3} + \frac{1}{3} \log_{2} (y)$

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

$\log_{2} (p) + \frac{\log_{2} (x)}{3} + \frac{\log_{2} (y)}{3}$