Precalculus Examples

$\log_{3} (\frac{\sqrt{x^{2} - 6}}{\sqrt[5]{2 + x^{2}}})$

Step 1

Rewrite $\log_{3} (\frac{\sqrt{x^{2} - 6}}{\sqrt[5]{2 + x^{2}}})$ as $\log_{3} (\sqrt{x^{2} - 6}) - \log_{3} (\sqrt[5]{2 + x^{2}})$ .

$\log_{3} (\sqrt{x^{2} - 6}) - \log_{3} (\sqrt[5]{2 + x^{2}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 6}$ as ${(x^{2} - 6)}^{\frac{1}{2}}$ .

$\log_{3} ({(x^{2} - 6)}^{\frac{1}{2}}) - \log_{3} (\sqrt[5]{2 + x^{2}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2 + x^{2}}$ as ${(2 + x^{2})}^{\frac{1}{5}}$ .

$\log_{3} ({(x^{2} - 6)}^{\frac{1}{2}}) - \log_{3} ({(2 + x^{2})}^{\frac{1}{5}})$

Step 4

Expand $\log_{3} ({(x^{2} - 6)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{3} (x^{2} - 6) - \log_{3} ({(2 + x^{2})}^{\frac{1}{5}})$

Step 5

Expand $\log_{3} ({(2 + x^{2})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{2} \log_{3} (x^{2} - 6) - (\frac{1}{5} \log_{3} (2 + x^{2}))$

Step 6

Combine $\frac{1}{5}$ and $\log_{3} (2 + x^{2})$ .

$\frac{1}{2} \log_{3} (x^{2} - 6) - \frac{\log_{3} (2 + x^{2})}{5}$

Step 7

Combine $\frac{1}{2}$ and $\log_{3} (x^{2} - 6)$ .

$\frac{\log_{3} (x^{2} - 6)}{2} - \frac{\log_{3} (2 + x^{2})}{5}$

Step 8

To write $\frac{\log_{3} (x^{2} - 6)}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\log_{3} (x^{2} - 6)}{2} \cdot \frac{5}{5} - \frac{\log_{3} (2 + x^{2})}{5}$

Step 9

To write $- \frac{\log_{3} (2 + x^{2})}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log_{3} (x^{2} - 6)}{2} \cdot \frac{5}{5} - \frac{\log_{3} (2 + x^{2})}{5} \cdot \frac{2}{2}$

Step 10

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{\log_{3} (x^{2} - 6)}{2}$ by $\frac{5}{5}$ .

$\frac{\log_{3} (x^{2} - 6) \cdot 5}{2 \cdot 5} - \frac{\log_{3} (2 + x^{2})}{5} \cdot \frac{2}{2}$

Step 10.2

Multiply $2$ by $5$ .

$\frac{\log_{3} (x^{2} - 6) \cdot 5}{10} - \frac{\log_{3} (2 + x^{2})}{5} \cdot \frac{2}{2}$

Step 10.3

Multiply $\frac{\log_{3} (2 + x^{2})}{5}$ by $\frac{2}{2}$ .

$\frac{\log_{3} (x^{2} - 6) \cdot 5}{10} - \frac{\log_{3} (2 + x^{2}) \cdot 2}{5 \cdot 2}$

Step 10.4

Multiply $5$ by $2$ .

$\frac{\log_{3} (x^{2} - 6) \cdot 5}{10} - \frac{\log_{3} (2 + x^{2}) \cdot 2}{10}$

Step 11

Combine the numerators over the common denominator.

$\frac{\log_{3} (x^{2} - 6) \cdot 5 - \log_{3} (2 + x^{2}) \cdot 2}{10}$

Step 12

Simplify the numerator.

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Step 12.1

Move $5$ to the left of $\log_{3} (x^{2} - 6)$ .

$\frac{5 \cdot \log_{3} (x^{2} - 6) - \log_{3} (2 + x^{2}) \cdot 2}{10}$

Step 12.2

Multiply $2$ by $- 1$ .

$\frac{5 \log_{3} (x^{2} - 6) - 2 \log_{3} (2 + x^{2})}{10}$