Precalculus Examples

$\log_{9} (\frac{x (x^{2} + 7)}{\sqrt{x^{2} - 7}})$

Step 1

Rewrite $\log_{9} (\frac{x (x^{2} + 7)}{\sqrt{x^{2} - 7}})$ as $\log_{9} (x (x^{2} + 7)) - \log_{9} (\sqrt{x^{2} - 7})$ .

$\log_{9} (x (x^{2} + 7)) - \log_{9} (\sqrt{x^{2} - 7})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 7}$ as ${(x^{2} - 7)}^{\frac{1}{2}}$ .

$\log_{9} (x (x^{2} + 7)) - \log_{9} ({(x^{2} - 7)}^{\frac{1}{2}})$

Step 3

Expand $\log_{9} ({(x^{2} - 7)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{9} (x (x^{2} + 7)) - (\frac{1}{2} \log_{9} (x^{2} - 7))$

Step 4

Combine $\frac{1}{2}$ and $\log_{9} (x^{2} - 7)$ .

$\log_{9} (x (x^{2} + 7)) - \frac{\log_{9} (x^{2} - 7)}{2}$

Step 5

Rewrite $\log_{9} (x (x^{2} + 7))$ as $\log_{9} (x) + \log_{9} (x^{2} + 7)$ .

$\log_{9} (x) + \log_{9} (x^{2} + 7) - \frac{\log_{9} (x^{2} - 7)}{2}$

Step 6

To write $\log_{9} (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{9} (x) \cdot \frac{2}{2} - \frac{\log_{9} (x^{2} - 7)}{2} + \log_{9} (x^{2} + 7)$

Step 7

Combine $\log_{9} (x)$ and $\frac{2}{2}$ .

$\frac{\log_{9} (x) \cdot 2}{2} - \frac{\log_{9} (x^{2} - 7)}{2} + \log_{9} (x^{2} + 7)$

Step 8

Combine the numerators over the common denominator.

$\frac{\log_{9} (x) \cdot 2 - \log_{9} (x^{2} - 7)}{2} + \log_{9} (x^{2} + 7)$

Step 9

Move $2$ to the left of $\log_{9} (x)$ .

$\frac{2 \log_{9} (x) - \log_{9} (x^{2} - 7)}{2} + \log_{9} (x^{2} + 7)$

Step 10

To write $\log_{9} (x^{2} + 7)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 \log_{9} (x) - \log_{9} (x^{2} - 7)}{2} + \log_{9} (x^{2} + 7) \cdot \frac{2}{2}$

Step 11

Combine $\log_{9} (x^{2} + 7)$ and $\frac{2}{2}$ .

$\frac{2 \log_{9} (x) - \log_{9} (x^{2} - 7)}{2} + \frac{\log_{9} (x^{2} + 7) \cdot 2}{2}$

Step 12

Combine the numerators over the common denominator.

$\frac{2 \log_{9} (x) - \log_{9} (x^{2} - 7) + \log_{9} (x^{2} + 7) \cdot 2}{2}$

Step 13

Move $2$ to the left of $\log_{9} (x^{2} + 7)$ .

$\frac{2 \log_{9} (x) - \log_{9} (x^{2} - 7) + 2 \log_{9} (x^{2} + 7)}{2}$