Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Divide $2$ by $2$ .

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

Step 4

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

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

Step 5

Simplify.

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