Expand the Logarithmic Expression log base 4 of ( cube root of xy)/64

$\log_{4} (\frac{\sqrt[3]{x y}}{64})$

Step 1

Rewrite $\log_{4} (\frac{\sqrt[3]{x y}}{64})$ as $\log_{4} (\sqrt[3]{x y}) - \log_{4} (64)$ .

$\log_{4} (\sqrt[3]{x y}) - \log_{4} (64)$

Step 2

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

$\log_{4} ({(x y)}^{\frac{1}{3}}) - \log_{4} (64)$

Step 3

Expand $\log_{4} ({(x y)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{4} (x y) - \log_{4} (64)$

Step 4

Logarithm base $4$ of $64$ is $3$ .

$\frac{1}{3} \log_{4} (x y) - 1 \cdot 3$

Step 5

Multiply $- 1$ by $3$ .

$\frac{1}{3} \log_{4} (x y) - 3$

Step 6

Rewrite $\log_{4} (x y)$ as $\log_{4} (x) + \log_{4} (y)$ .

$\frac{1}{3} (\log_{4} (x) + \log_{4} (y)) - 3$

Step 7

Simplify each term.

Apply the distributive property.

$\frac{1}{3} \log_{4} (x) + \frac{1}{3} \log_{4} (y) - 3$

Combine $\frac{1}{3}$ and $\log_{4} (x)$ .

$\frac{\log_{4} (x)}{3} + \frac{1}{3} \log_{4} (y) - 3$

Combine $\frac{1}{3}$ and $\log_{4} (y)$ .

$\frac{\log_{4} (x)}{3} + \frac{\log_{4} (y)}{3} - 3$