Expand the Logarithmic Expression log base 2 of (4x^7y^2)/(12z)

$\log_{2} (\frac{4 x^{7} y^{2}}{12 z})$

Step 1

Rewrite $\log_{2} (\frac{4 x^{7} y^{2}}{12 z})$ as $\log_{2} (4 x^{7} y^{2}) - \log_{2} (12 z)$ .

$\log_{2} (4 x^{7} y^{2}) - \log_{2} (12 z)$

Step 2

Rewrite $\log_{2} (4 x^{7} y^{2})$ as $\log_{2} (4 x^{7}) + \log_{2} (y^{2})$ .

$\log_{2} (4 x^{7}) + \log_{2} (y^{2}) - \log_{2} (12 z)$

Step 3

Rewrite $\log_{2} (4 x^{7})$ as $\log_{2} (4) + \log_{2} (x^{7})$ .

$\log_{2} (4) + \log_{2} (x^{7}) + \log_{2} (y^{2}) - \log_{2} (12 z)$

Step 4

Expand $\log_{2} (x^{7})$ by moving $7$ outside the logarithm.

$\log_{2} (4) + 7 \log_{2} (x) + \log_{2} (y^{2}) - \log_{2} (12 z)$

Step 5

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

$\log_{2} (4) + 7 \log_{2} (x) + 2 \log_{2} (y) - \log_{2} (12 z)$

Step 6

Logarithm base $2$ of $4$ is $2$ .

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - \log_{2} (12 z)$

Step 7

Rewrite $\log_{2} (12 z)$ as $\log_{2} (12) + \log_{2} (z)$ .

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - (\log_{2} (12) + \log_{2} (z))$

Step 8

Simplify each term.

Rewrite $\log_{2} (12)$ as $\log_{2} (2^{2} \cdot 3)$ .

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - (\log_{2} (2^{2} \cdot 3) + \log_{2} (z))$

Rewrite $\log_{2} (2^{2} \cdot 3)$ as $\log_{2} (2^{2}) + \log_{2} (3)$ .

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - (\log_{2} (2^{2}) + \log_{2} (3) + \log_{2} (z))$

Use logarithm rules to move $2$ out of the exponent.

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - (2 \log_{2} (2) + \log_{2} (3) + \log_{2} (z))$

Logarithm base $2$ of $2$ is $1$ .

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - (2 \cdot 1 + \log_{2} (3) + \log_{2} (z))$

Multiply $2$ by $1$ .

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - (2 + \log_{2} (3) + \log_{2} (z))$

Apply the distributive property.

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - 1 \cdot 2 - \log_{2} (3) - \log_{2} (z)$

Multiply $- 1$ by $2$ .

$2 + 7 \log_{2} (x) + 2 \log_{2} (y) - 2 - \log_{2} (3) - \log_{2} (z)$

Step 9

Subtract $2$ from $2$ .

$7 \log_{2} (x) + 2 \log_{2} (y) + 0 - \log_{2} (3) - \log_{2} (z)$

Step 10

Add $7 \log_{2} (x)$ and $0$ .

$7 \log_{2} (x) + 2 \log_{2} (y) - \log_{2} (3) - \log_{2} (z)$