Calculus Examples

$\log_{8} (320) - \log_{8} (5)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log_{8} (320)$ as $\log_{8} (2^{6} \cdot 5)$ .

$\log_{8} (2^{6} \cdot 5) - \log_{8} (5)$

Step 1.2

Rewrite $\log_{8} (2^{6} \cdot 5)$ as $\log_{8} (2^{6}) + \log_{8} (5)$ .

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

Step 1.3

Simplify each term.

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Step 1.3.1

Expand $\log_{8} (2^{6})$ by moving $6$ outside the logarithm.

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

Step 1.3.2

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

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

Step 1.3.3

Cancel the common factor of $3$ .

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Step 1.3.3.1

Factor $3$ out of $6$ .

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

Step 1.3.3.2

Cancel the common factor.

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

Step 1.3.3.3

Rewrite the expression.

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

Step 2

Subtract $\log_{8} (5)$ from $\log_{8} (5)$ .

$2 + 0$

Step 3

Add $2$ and $0$ .

$2$