Precalculus Examples

$\log_{8} (\sqrt{\frac{13}{10}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{13}{10}}$ as ${(\frac{13}{10})}^{\frac{1}{2}}$ .

$\log_{8} ({(\frac{13}{10})}^{\frac{1}{2}})$

Step 2

Expand $\log_{8} ({(\frac{13}{10})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{8} (\frac{13}{10})$

Step 3

Combine $\frac{1}{2}$ and $\log_{8} (\frac{13}{10})$ .

$\frac{\log_{8} (\frac{13}{10})}{2}$

Step 4

Rewrite $\log_{8} (\frac{13}{10})$ as $\log_{8} (13) - \log_{8} (10)$ .

$\frac{\log_{8} (13) - \log_{8} (10)}{2}$

Step 5

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

$\frac{\log_{8} (13) - \log_{8} (2 (5))}{2}$

Step 6

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

$\frac{\log_{8} (13) - (\log_{8} (2) + \log_{8} (5))}{2}$

Step 7

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

$\frac{\log_{8} (13) - (\frac{1}{3} + \log_{8} (5))}{2}$

Step 8

Apply the distributive property.

$\frac{\log_{8} (13) - \frac{1}{3} - \log_{8} (5)}{2}$

Step 9

Simplify the numerator.

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Step 9.1

To write $\log_{8} (13)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\log_{8} (13) \cdot \frac{3}{3} - \frac{1}{3} - \log_{8} (5)}{2}$

Step 9.2

Combine $\log_{8} (13)$ and $\frac{3}{3}$ .

$\frac{\frac{\log_{8} (13) \cdot 3}{3} - \frac{1}{3} - \log_{8} (5)}{2}$

Step 9.3

Combine the numerators over the common denominator.

$\frac{\frac{\log_{8} (13) \cdot 3 - 1}{3} - \log_{8} (5)}{2}$

Step 9.4

Move $3$ to the left of $\log_{8} (13)$ .

$\frac{\frac{3 \log_{8} (13) - 1}{3} - \log_{8} (5)}{2}$

Step 9.5

To write $- \log_{8} (5)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{3 \log_{8} (13) - 1}{3} - \log_{8} (5) \cdot \frac{3}{3}}{2}$

Step 9.6

Combine $- \log_{8} (5)$ and $\frac{3}{3}$ .

$\frac{\frac{3 \log_{8} (13) - 1}{3} + \frac{- \log_{8} (5) \cdot 3}{3}}{2}$

Step 9.7

Combine the numerators over the common denominator.

$\frac{\frac{3 \log_{8} (13) - 1 - \log_{8} (5) \cdot 3}{3}}{2}$

Step 9.8

Multiply $3$ by $- 1$ .

$\frac{\frac{3 \log_{8} (13) - 1 - 3 \log_{8} (5)}{3}}{2}$

Step 10

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 \log_{8} (13) - 1 - 3 \log_{8} (5)}{3} \cdot \frac{1}{2}$

Step 11

Multiply $\frac{3 \log_{8} (13) - 1 - 3 \log_{8} (5)}{3} \cdot \frac{1}{2}$ .

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Step 11.1

Multiply $\frac{3 \log_{8} (13) - 1 - 3 \log_{8} (5)}{3}$ by $\frac{1}{2}$ .

$\frac{3 \log_{8} (13) - 1 - 3 \log_{8} (5)}{3 \cdot 2}$

Step 11.2

Multiply $3$ by $2$ .

$\frac{3 \log_{8} (13) - 1 - 3 \log_{8} (5)}{6}$