Precalculus Examples

$\log_{8} (\frac{\sqrt[5]{x^{2} - 3}}{\sqrt[4]{9 + x^{2}}})$

Step 1

Rewrite $\log_{8} (\frac{\sqrt[5]{x^{2} - 3}}{\sqrt[4]{9 + x^{2}}})$ as $\log_{8} (\sqrt[5]{x^{2} - 3}) - \log_{8} (\sqrt[4]{9 + x^{2}})$ .

$\log_{8} (\sqrt[5]{x^{2} - 3}) - \log_{8} (\sqrt[4]{9 + x^{2}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2} - 3}$ as ${(x^{2} - 3)}^{\frac{1}{5}}$ .

$\log_{8} ({(x^{2} - 3)}^{\frac{1}{5}}) - \log_{8} (\sqrt[4]{9 + x^{2}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{9 + x^{2}}$ as ${(9 + x^{2})}^{\frac{1}{4}}$ .

$\log_{8} ({(x^{2} - 3)}^{\frac{1}{5}}) - \log_{8} ({(9 + x^{2})}^{\frac{1}{4}})$

Step 4

Expand $\log_{8} ({(x^{2} - 3)}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log_{8} (x^{2} - 3) - \log_{8} ({(9 + x^{2})}^{\frac{1}{4}})$

Step 5

Expand $\log_{8} ({(9 + x^{2})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{5} \log_{8} (x^{2} - 3) - (\frac{1}{4} \log_{8} (9 + x^{2}))$

Step 6

Combine $\frac{1}{4}$ and $\log_{8} (9 + x^{2})$ .

$\frac{1}{5} \log_{8} (x^{2} - 3) - \frac{\log_{8} (9 + x^{2})}{4}$

Step 7

Combine $\frac{1}{5}$ and $\log_{8} (x^{2} - 3)$ .

$\frac{\log_{8} (x^{2} - 3)}{5} - \frac{\log_{8} (9 + x^{2})}{4}$

Step 8

To write $\frac{\log_{8} (x^{2} - 3)}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\log_{8} (x^{2} - 3)}{5} \cdot \frac{4}{4} - \frac{\log_{8} (9 + x^{2})}{4}$

Step 9

To write $- \frac{\log_{8} (9 + x^{2})}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\log_{8} (x^{2} - 3)}{5} \cdot \frac{4}{4} - \frac{\log_{8} (9 + x^{2})}{4} \cdot \frac{5}{5}$

Step 10

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{\log_{8} (x^{2} - 3)}{5}$ by $\frac{4}{4}$ .

$\frac{\log_{8} (x^{2} - 3) \cdot 4}{5 \cdot 4} - \frac{\log_{8} (9 + x^{2})}{4} \cdot \frac{5}{5}$

Step 10.2

Multiply $5$ by $4$ .

$\frac{\log_{8} (x^{2} - 3) \cdot 4}{20} - \frac{\log_{8} (9 + x^{2})}{4} \cdot \frac{5}{5}$

Step 10.3

Multiply $\frac{\log_{8} (9 + x^{2})}{4}$ by $\frac{5}{5}$ .

$\frac{\log_{8} (x^{2} - 3) \cdot 4}{20} - \frac{\log_{8} (9 + x^{2}) \cdot 5}{4 \cdot 5}$

Step 10.4

Multiply $4$ by $5$ .

$\frac{\log_{8} (x^{2} - 3) \cdot 4}{20} - \frac{\log_{8} (9 + x^{2}) \cdot 5}{20}$

Step 11

Combine the numerators over the common denominator.

$\frac{\log_{8} (x^{2} - 3) \cdot 4 - \log_{8} (9 + x^{2}) \cdot 5}{20}$

Step 12

Simplify the numerator.

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Step 12.1

Move $4$ to the left of $\log_{8} (x^{2} - 3)$ .

$\frac{4 \cdot \log_{8} (x^{2} - 3) - \log_{8} (9 + x^{2}) \cdot 5}{20}$

Step 12.2

Multiply $5$ by $- 1$ .

$\frac{4 \log_{8} (x^{2} - 3) - 5 \log_{8} (9 + x^{2})}{20}$