Precalculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x - 3}{x + 3}}$ as ${(\frac{x - 3}{x + 3})}^{\frac{1}{2}}$ .

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

Step 2

Expand $\log_{2} ({(\frac{x - 3}{x + 3})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 3

Rewrite $\log_{2} (\frac{x - 3}{x + 3})$ as $\log_{2} (x - 3) - \log_{2} (x + 3)$ .

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

Step 4

Apply the distributive property.

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

Step 5

Combine $\frac{1}{2}$ and $\log_{2} (x - 3)$ .

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

Step 6

Combine $\frac{1}{2}$ and $\log_{2} (x + 3)$ .

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

Step 7

Combine the numerators over the common denominator.

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