Precalculus Examples

$\log_{8} (6 \sqrt{x y})$

Step 1

Rewrite $\log_{8} (6 \sqrt{x y})$ as $\log_{8} (6) + \log_{8} (\sqrt{x y})$ .

$\log_{8} (6) + \log_{8} (\sqrt{x y})$

Step 2

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

$\log_{8} (6) + \log_{8} ({(x y)}^{\frac{1}{2}})$

Step 3

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

$\log_{8} (6) + \frac{1}{2} \log_{8} (x y)$

Step 4

Rewrite $\log_{8} (x y)$ as $\log_{8} (x) + \log_{8} (y)$ .

$\log_{8} (6) + \frac{1}{2} (\log_{8} (x) + \log_{8} (y))$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\log_{8} (6)$ as $\log_{8} (2 (3))$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Apply the distributive property.

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

Step 5.5

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

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

Step 5.6

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

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