Precalculus Examples

$\log_{8} (11^{5} \sqrt{10})$

Step 1

Rewrite $\log_{8} (11^{5} \sqrt{10})$ as $\log_{8} (11^{5}) + \log_{8} (\sqrt{10})$ .

$\log_{8} (11^{5}) + \log_{8} (\sqrt{10})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\log_{8} (11^{5}) + \log_{8} (10^{\frac{1}{2}})$

Step 3

Expand $\log_{8} (11^{5})$ by moving $5$ outside the logarithm.

$5 \log_{8} (11) + \log_{8} (10^{\frac{1}{2}})$

Step 4

Expand $\log_{8} (10^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$5 \log_{8} (11) + \frac{1}{2} \log_{8} (10)$

Step 5

Rewrite $\log_{8} (10)$ as $\log_{8} (2 (5))$ .

$5 \log_{8} (11) + \frac{1}{2} \log_{8} (2 (5))$

Step 6

Rewrite $\log_{8} (2 (5))$ as $\log_{8} (2) + \log_{8} (5)$ .

$5 \log_{8} (11) + \frac{1}{2} (\log_{8} (2) + \log_{8} (5))$

Step 7

Logarithm base $8$ of $2$ is $\frac{1}{3}$ .

$5 \log_{8} (11) + \frac{1}{2} (\frac{1}{3} + \log_{8} (5))$

Step 8

Apply the distributive property.

$5 \log_{8} (11) + \frac{1}{2} \cdot \frac{1}{3} + \frac{1}{2} \log_{8} (5)$

Step 9

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 9.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$5 \log_{8} (11) + \frac{1}{2 \cdot 3} + \frac{1}{2} \log_{8} (5)$

Step 9.2

Multiply $2$ by $3$ .

$5 \log_{8} (11) + \frac{1}{6} + \frac{1}{2} \log_{8} (5)$

Step 10

Combine $\frac{1}{2}$ and $\log_{8} (5)$ .

$5 \log_{8} (11) + \frac{1}{6} + \frac{\log_{8} (5)}{2}$