Precalculus Examples

$\log_{2} (\frac{\sqrt{a - 1}}{9})$

Step 1

Rewrite $\log_{2} (\frac{\sqrt{a - 1}}{9})$ as $\log_{2} (\sqrt{a - 1}) - \log_{2} (9)$ .

$\log_{2} (\sqrt{a - 1}) - \log_{2} (9)$

Step 2

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

$\log_{2} ({(a - 1)}^{\frac{1}{2}}) - \log_{2} (9)$

Step 3

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

$\frac{1}{2} \log_{2} (a - 1) - \log_{2} (9)$

Step 4

Rewrite $\log_{2} (9)$ as $\log_{2} (3^{2})$ .

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

Step 5

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

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

Step 6

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Multiply $2$ by $- 2$ .

$\frac{\log_{2} (a - 1) - 4 \log_{2} (3)}{2}$