Expand the Logarithmic Expression log base 4 of (10x^3y)/(z^2)

$\log_{4} (\frac{10 x^{3} y}{z^{2}})$

Step 1

Rewrite $\log_{4} (\frac{10 x^{3} y}{z^{2}})$ as $\log_{4} (10 x^{3} y) - \log_{4} (z^{2})$ .

$\log_{4} (10 x^{3} y) - \log_{4} (z^{2})$

Step 2

Expand $\log_{4} (z^{2})$ by moving $2$ outside the logarithm.

$\log_{4} (10 x^{3} y) - (2 \log_{4} (z))$

Step 3

Multiply $2$ by $- 1$ .

$\log_{4} (10 x^{3} y) - 2 \log_{4} (z)$

Step 4

Rewrite $\log_{4} (10 x^{3} y)$ as $\log_{4} (10 x^{3}) + \log_{4} (y)$ .

$\log_{4} (10 x^{3}) + \log_{4} (y) - 2 \log_{4} (z)$

Step 5

Rewrite $\log_{4} (10 x^{3})$ as $\log_{4} (10) + \log_{4} (x^{3})$ .

$\log_{4} (10) + \log_{4} (x^{3}) + \log_{4} (y) - 2 \log_{4} (z)$

Step 6

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

$\log_{4} (10) + 3 \log_{4} (x) + \log_{4} (y) - 2 \log_{4} (z)$

Step 7

Simplify each term.

Rewrite $\log_{4} (10)$ as $\log_{4} (2 (5))$ .

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

Rewrite $\log_{4} (2 (5))$ as $\log_{4} (2) + \log_{4} (5)$ .

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

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

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