Calculus Examples

$\log_{8} (\frac{\sqrt{10}}{9})$

Step 1

Rewrite $\log_{8} (\frac{\sqrt{10}}{9})$ as $\log_{8} (\sqrt{10}) - \log_{8} (9)$ .

$\log_{8} (\sqrt{10}) - \log_{8} (9)$

Step 2

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

$\log_{8} (10^{\frac{1}{2}}) - \log_{8} (9)$

Step 3

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

$\frac{1}{2} \log_{8} (10) - \log_{8} (9)$

Step 4

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

$\frac{1}{2} \log_{8} (2 (5)) - \log_{8} (9)$

Step 5

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

$\frac{1}{2} (\log_{8} (2) + \log_{8} (5)) - \log_{8} (9)$

Step 6

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

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

Step 7

Apply the distributive property.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Multiply $2$ by $3$ .

$\frac{1}{6} + \frac{1}{2} \log_{8} (5) - \log_{8} (9)$

Step 9

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

$\frac{1}{6} + \frac{\log_{8} (5)}{2} - \log_{8} (9)$

Step 10

Rewrite $\log_{8} (9)$ as $\log_{8} (3^{2})$ .

$\frac{1}{6} + \frac{\log_{8} (5)}{2} - \log_{8} (3^{2})$

Step 11

Expand $\log_{8} (3^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{6} + \frac{\log_{8} (5)}{2} - (2 \log_{8} (3))$

Step 12

Multiply $2$ by $- 1$ .

$\frac{1}{6} + \frac{\log_{8} (5)}{2} - 2 \log_{8} (3)$