Precalculus Examples

$\log_{8} ({(\frac{6}{7})}^{3})$

Step 1

Expand $\log_{8} ({(\frac{6}{7})}^{3})$ by moving $3$ outside the logarithm.

$3 \log_{8} (\frac{6}{7})$

Step 2

Rewrite $\log_{8} (\frac{6}{7})$ as $\log_{8} (6) - \log_{8} (7)$ .

$3 (\log_{8} (6) - \log_{8} (7))$

Step 3

Simplify each term.

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Step 3.1

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

$3 (\log_{8} (2 (3)) - \log_{8} (7))$

Step 3.2

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

$3 (\log_{8} (2) + \log_{8} (3) - \log_{8} (7))$

Step 3.3

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

$3 (\frac{1}{3} + \log_{8} (3) - \log_{8} (7))$

Step 4

Apply the distributive property.

$3 (\frac{1}{3}) + 3 \log_{8} (3) + 3 (- \log_{8} (7))$

Step 5

Simplify.

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Step 5.1

Cancel the common factor of $3$ .

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Step 5.1.1

Cancel the common factor.

$3 (\frac{1}{3}) + 3 \log_{8} (3) + 3 (- \log_{8} (7))$

Step 5.1.2

Rewrite the expression.

$1 + 3 \log_{8} (3) + 3 (- \log_{8} (7))$

Step 5.2

Multiply $- 1$ by $3$ .

$1 + 3 \log_{8} (3) - 3 \log_{8} (7)$