Precalculus Examples

$\log_{2} (\frac{\sqrt{a - 1}}{3})$

Step 1

Rewrite $\log_{2} (\frac{\sqrt{a - 1}}{3})$ as $\log_{2} (\sqrt{a - 1}) - \log_{2} (3)$ .

$\log_{2} (\sqrt{a - 1}) - \log_{2} (3)$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

To write $- \log_{2} (3)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log_{2} (a - 1)}{2} - \log_{2} (3) \cdot \frac{2}{2}$

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

$\frac{\log_{2} (a - 1) - \log_{2} (3) \cdot 2}{2}$

Step 8

Multiply $2$ by $- 1$ .

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