Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Move $2$ to the left of $\log_{2} (x)$ .

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

Step 10

To write $\log_{2} (x^{2} + 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Move $2$ to the left of $\log_{2} (x^{2} + 1)$ .

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