Precalculus Examples

$\log_{8} (\frac{18}{11})$

Step 1

Rewrite $\log_{8} (\frac{18}{11})$ as $\log_{8} (18) - \log_{8} (11)$ .

$\log_{8} (18) - \log_{8} (11)$

Step 2

Simplify each term.

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Step 2.1

Rewrite $\log_{8} (18)$ as $\log_{8} (2 \cdot 3^{2})$ .

$\log_{8} (2 \cdot 3^{2}) - \log_{8} (11)$

Step 2.2

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

$\log_{8} (2) + \log_{8} (3^{2}) - \log_{8} (11)$

Step 2.3

Simplify each term.

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Step 2.3.1

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

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

Step 2.3.2

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

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