Precalculus Examples

$\log_{8} (5^{5} \sqrt{12})$

Step 1

Rewrite $\log_{8} (5^{5} \sqrt{12})$ as $\log_{8} (5^{5}) + \log_{8} (\sqrt{12})$ .

$\log_{8} (5^{5}) + \log_{8} (\sqrt{12})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12}$ as $12^{\frac{1}{2}}$ .

$\log_{8} (5^{5}) + \log_{8} (12^{\frac{1}{2}})$

Step 3

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

$5 \log_{8} (5) + \log_{8} (12^{\frac{1}{2}})$

Step 4

Expand $\log_{8} (12^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$5 \log_{8} (5) + \frac{1}{2} \log_{8} (12)$

Step 5

Rewrite $\log_{8} (12)$ as $\log_{8} (2^{2} \cdot 3)$ .

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

Step 6

Rewrite $\log_{8} (2^{2} \cdot 3)$ as $\log_{8} (2^{2}) + \log_{8} (3)$ .

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

Step 7

Simplify each term.

Tap for more steps...

Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Combine $2$ and $\frac{1}{3}$ .

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

Step 8

Apply the distributive property.

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

Step 9

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

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

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