Precalculus Examples

$\log_{8} (\sqrt{7 \cdot 3 \cdot 11})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 \cdot 3 \cdot 11}$ as ${(7 \cdot 3 \cdot 11)}^{\frac{1}{2}}$ .

$\log_{8} ({(7 \cdot 3 \cdot 11)}^{\frac{1}{2}})$

Step 2

Expand $\log_{8} ({(7 \cdot 3 \cdot 11)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{8} (7 \cdot 3 \cdot 11)$

Step 3

Multiply $7$ by $3$ .

$\frac{1}{2} \log_{8} (21 \cdot 11)$

Step 4

Multiply $21$ by $11$ .

$\frac{1}{2} \log_{8} (231)$

Step 5

Rewrite $\log_{8} (231)$ as $\log_{8} (3 \cdot 7 \cdot 11)$ .

$\frac{1}{2} \log_{8} (3 \cdot 7 \cdot 11)$

Step 6

Rewrite $\log_{8} (3 \cdot 7 \cdot 11)$ as $\log_{8} (3 \cdot 7) + \log_{8} (11)$ .

$\frac{1}{2} (\log_{8} (3 \cdot 7) + \log_{8} (11))$

Step 7

Multiply $3$ by $7$ .

$\frac{1}{2} (\log_{8} (21) + \log_{8} (11))$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\log_{8} (21)$ as $\log_{8} (3 (7))$ .

$\frac{1}{2} (\log_{8} (3 (7)) + \log_{8} (11))$

Step 8.2

Rewrite $\log_{8} (3 (7))$ as $\log_{8} (3) + \log_{8} (7)$ .

$\frac{1}{2} (\log_{8} (3) + \log_{8} (7) + \log_{8} (11))$

Step 9

Apply the distributive property.

$\frac{1}{2} \log_{8} (3) + \frac{1}{2} \log_{8} (7) + \frac{1}{2} \log_{8} (11)$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and $\log_{8} (3)$ .

$\frac{\log_{8} (3)}{2} + \frac{1}{2} \log_{8} (7) + \frac{1}{2} \log_{8} (11)$

Step 10.2

Combine $\frac{1}{2}$ and $\log_{8} (7)$ .

$\frac{\log_{8} (3)}{2} + \frac{\log_{8} (7)}{2} + \frac{1}{2} \log_{8} (11)$

Step 10.3

Combine $\frac{1}{2}$ and $\log_{8} (11)$ .

$\frac{\log_{8} (3)}{2} + \frac{\log_{8} (7)}{2} + \frac{\log_{8} (11)}{2}$