Expand the Logarithmic Expression log base 4 of (5x^(3y))/(2xy^3)

$\log_{4} (\frac{5 x^{3 y}}{2 x y^{3}})$

Rewrite $\log_{4} (\frac{5 x^{3 y}}{2 x y^{3}})$ as $\log_{4} (5 x^{3 y}) - \log_{4} (2 x y^{3})$ .

$\log_{4} (5 x^{3 y}) - \log_{4} (2 x y^{3})$

Rewrite $\log_{4} (5 x^{3 y})$ as $\log_{4} (5) + \log_{4} (x^{3 y})$ .

$\log_{4} (5) + \log_{4} (x^{3 y}) - \log_{4} (2 x y^{3})$

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

$\log_{4} (5) + 3 y \log_{4} (x) - \log_{4} (2 x y^{3})$

Rewrite $\log_{4} (2 x y^{3})$ as $\log_{4} (2 x) + \log_{4} (y^{3})$ .

$\log_{4} (5) + 3 y \log_{4} (x) - (\log_{4} (2 x) + \log_{4} (y^{3}))$

Rewrite $\log_{4} (2 x)$ as $\log_{4} (2) + \log_{4} (x)$ .

$\log_{4} (5) + 3 y \log_{4} (x) - (\log_{4} (2) + \log_{4} (x) + \log_{4} (y^{3}))$

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

$\log_{4} (5) + 3 y \log_{4} (x) - (\log_{4} (2) + \log_{4} (x) + 3 \log_{4} (y))$

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\log_{4} (5) + 3 y \log_{4} (x) - (\frac{1}{2} + \log_{4} (x) + 3 \log_{4} (y))$

Simplify each term.

Apply the distributive property.

$\log_{4} (5) + 3 y \log_{4} (x) - \frac{1}{2} - \log_{4} (x) - (3 \log_{4} (y))$

Multiply $3$ by $- 1$ .

$\log_{4} (5) + 3 y \log_{4} (x) - \frac{1}{2} - \log_{4} (x) - 3 \log_{4} (y)$