Algebra Examples

$\log_{2} (\frac{9 x^{5}}{4 y^{3}})$

Step 1

Rewrite $\log_{2} (\frac{9 x^{5}}{4 y^{3}})$ as $\log_{2} (9 x^{5}) - \log_{2} (4 y^{3})$ .

$\log_{2} (9 x^{5}) - \log_{2} (4 y^{3})$

Step 2

Rewrite $\log_{2} (9 x^{5})$ as $\log_{2} (9) + \log_{2} (x^{5})$ .

$\log_{2} (9) + \log_{2} (x^{5}) - \log_{2} (4 y^{3})$

Step 3

Expand $\log_{2} (x^{5})$ by moving $5$ outside the logarithm.

$\log_{2} (9) + 5 \log_{2} (x) - \log_{2} (4 y^{3})$

Step 4

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

$\log_{2} (9) + 5 \log_{2} (x) - (\log_{2} (4) + \log_{2} (y^{3}))$

Step 5

Expand $\log_{2} (y^{3})$ by moving $3$ outside the logarithm.

$\log_{2} (9) + 5 \log_{2} (x) - (\log_{2} (4) + 3 \log_{2} (y))$

Step 6

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

$\log_{2} (9) + 5 \log_{2} (x) - (2 + 3 \log_{2} (y))$

Step 7

Simplify each term.

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Step 7.1

Rewrite $\log_{2} (9)$ as $\log_{2} (3^{2})$ .

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

Step 7.2

Expand $\log_{2} (3^{2})$ by moving $2$ outside the logarithm.

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Multiply $- 1$ by $2$ .

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

Step 7.5

Multiply $3$ by $- 1$ .

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