Expand the Logarithmic Expression log base 4 of (64x^3)/(y^4z^6)

$\log_{4} (\frac{64 x^{3}}{y^{4} z^{6}})$

Rewrite $\log_{4} (\frac{64 x^{3}}{y^{4} z^{6}})$ as $\log_{4} (64 x^{3}) - \log_{4} (y^{4} z^{6})$ .

$\log_{4} (64 x^{3}) - \log_{4} (y^{4} z^{6})$

Rewrite $\log_{4} (64 x^{3})$ as $\log_{4} (64) + \log_{4} (x^{3})$ .

$\log_{4} (64) + \log_{4} (x^{3}) - \log_{4} (y^{4} z^{6})$

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

$\log_{4} (64) + 3 \log_{4} (x) - \log_{4} (y^{4} z^{6})$

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

$3 + 3 \log_{4} (x) - \log_{4} (y^{4} z^{6})$

Rewrite $\log_{4} (y^{4} z^{6})$ as $\log_{4} (y^{4}) + \log_{4} (z^{6})$ .

$3 + 3 \log_{4} (x) - (\log_{4} (y^{4}) + \log_{4} (z^{6}))$

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

$3 + 3 \log_{4} (x) - (4 \log_{4} (y) + \log_{4} (z^{6}))$

Expand $\log_{4} (z^{6})$ by moving $6$ outside the logarithm.

$3 + 3 \log_{4} (x) - (4 \log_{4} (y) + 6 \log_{4} (z))$

Simplify each term.

Apply the distributive property.

$3 + 3 \log_{4} (x) - (4 \log_{4} (y)) - (6 \log_{4} (z))$

Multiply $4$ by $- 1$ .

$3 + 3 \log_{4} (x) - 4 \log_{4} (y) - (6 \log_{4} (z))$

Multiply $6$ by $- 1$ .

$3 + 3 \log_{4} (x) - 4 \log_{4} (y) - 6 \log_{4} (z)$