Precalculus Examples

$\log_{3} (\frac{\sqrt{x^{2} - 2}}{\sqrt[4]{7 + x^{2}}})$

Step 1

Rewrite $\log_{3} (\frac{\sqrt{x^{2} - 2}}{\sqrt[4]{7 + x^{2}}})$ as $\log_{3} (\sqrt{x^{2} - 2}) - \log_{3} (\sqrt[4]{7 + x^{2}})$ .

$\log_{3} (\sqrt{x^{2} - 2}) - \log_{3} (\sqrt[4]{7 + x^{2}})$

Step 2

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

$\log_{3} ({(x^{2} - 2)}^{\frac{1}{2}}) - \log_{3} (\sqrt[4]{7 + x^{2}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{7 + x^{2}}$ as ${(7 + x^{2})}^{\frac{1}{4}}$ .

$\log_{3} ({(x^{2} - 2)}^{\frac{1}{2}}) - \log_{3} ({(7 + x^{2})}^{\frac{1}{4}})$

Step 4

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

$\frac{1}{2} \log_{3} (x^{2} - 2) - \log_{3} ({(7 + x^{2})}^{\frac{1}{4}})$

Step 5

Expand $\log_{3} ({(7 + x^{2})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{2} \log_{3} (x^{2} - 2) - (\frac{1}{4} \log_{3} (7 + x^{2}))$

Step 6

Combine $\frac{1}{4}$ and $\log_{3} (7 + x^{2})$ .

$\frac{1}{2} \log_{3} (x^{2} - 2) - \frac{\log_{3} (7 + x^{2})}{4}$

Step 7

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

$\frac{\log_{3} (x^{2} - 2)}{2} - \frac{\log_{3} (7 + x^{2})}{4}$

Step 8

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

$\frac{\log_{3} (x^{2} - 2)}{2} \cdot \frac{2}{2} - \frac{\log_{3} (7 + x^{2})}{4}$

Step 9

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

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

$\frac{\log_{3} (x^{2} - 2) \cdot 2}{2 \cdot 2} - \frac{\log_{3} (7 + x^{2})}{4}$

Step 9.2

Multiply $2$ by $2$ .

$\frac{\log_{3} (x^{2} - 2) \cdot 2}{4} - \frac{\log_{3} (7 + x^{2})}{4}$

Step 10

Combine the numerators over the common denominator.

$\frac{\log_{3} (x^{2} - 2) \cdot 2 - \log_{3} (7 + x^{2})}{4}$

Step 11

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

$\frac{2 \log_{3} (x^{2} - 2) - \log_{3} (7 + x^{2})}{4}$