Precalculus Examples

$\log_{5} (\sqrt{\frac{x - 4}{x + 4}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x - 4}{x + 4}}$ as ${(\frac{x - 4}{x + 4})}^{\frac{1}{2}}$ .

$\log_{5} ({(\frac{x - 4}{x + 4})}^{\frac{1}{2}})$

Step 2

Expand $\log_{5} ({(\frac{x - 4}{x + 4})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{5} (\frac{x - 4}{x + 4})$

Step 3

Rewrite $\log_{5} (\frac{x - 4}{x + 4})$ as $\log_{5} (x - 4) - \log_{5} (x + 4)$ .

$\frac{1}{2} (\log_{5} (x - 4) - \log_{5} (x + 4))$

Step 4

Apply the distributive property.

$\frac{1}{2} \log_{5} (x - 4) + \frac{1}{2} (- \log_{5} (x + 4))$

Step 5

Combine $\frac{1}{2}$ and $\log_{5} (x - 4)$ .

$\frac{\log_{5} (x - 4)}{2} + \frac{1}{2} (- \log_{5} (x + 4))$

Step 6

Combine $\frac{1}{2}$ and $\log_{5} (x + 4)$ .

$\frac{\log_{5} (x - 4)}{2} - \frac{\log_{5} (x + 4)}{2}$

Step 7

Combine the numerators over the common denominator.

$\frac{\log_{5} (x - 4) - \log_{5} (x + 4)}{2}$