Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\log_{2} (8 x^{\frac{1}{2}}) - \log_{2} (y^{2})$

Step 3

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

$\log_{2} (8 x^{\frac{1}{2}}) - (2 \log_{2} (y))$

Step 4

Multiply $2$ by $- 1$ .

$\log_{2} (8 x^{\frac{1}{2}}) - 2 \log_{2} (y)$

Step 5

Rewrite $\log_{2} (8 x^{\frac{1}{2}})$ as $\log_{2} (8) + \log_{2} (x^{\frac{1}{2}})$ .

$\log_{2} (8) + \log_{2} (x^{\frac{1}{2}}) - 2 \log_{2} (y)$

Step 6

Expand $\log_{2} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{2} (8) + \frac{1}{2} \log_{2} (x) - 2 \log_{2} (y)$

Step 7

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

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

Step 8

Combine $\frac{1}{2}$ and $\log_{2} (x)$ .

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