Precalculus Examples

$\log_{2} (\frac{8 x^{3}}{2 y})$

Step 1

Rewrite $\log_{2} (\frac{8 x^{3}}{2 y})$ as $\log_{2} (8 x^{3}) - \log_{2} (2 y)$ .

$\log_{2} (8 x^{3}) - \log_{2} (2 y)$

Step 2

Rewrite $\log_{2} (8 x^{3})$ as $\log_{2} (8) + \log_{2} (x^{3})$ .

$\log_{2} (8) + \log_{2} (x^{3}) - \log_{2} (2 y)$

Step 3

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

$\log_{2} (8) + 3 \log_{2} (x) - \log_{2} (2 y)$

Step 4

Logarithm base $2$ of $8$ is $3$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $- 1$ by $1$ .

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

Step 8

Subtract $1$ from $3$ .

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