Calculus Examples

$\log_{8} (\sqrt{80}) - \log_{8} (\sqrt{5})$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{8} (\frac{\sqrt{80}}{\sqrt{5}})$

Step 2

Combine $\sqrt{80}$ and $\sqrt{5}$ into a single radical.

$\log_{8} (\sqrt{\frac{80}{5}})$

Step 3

Simplify the expression.

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Step 3.1

Divide $80$ by $5$ .

$\log_{8} (\sqrt{16})$

Step 3.2

Rewrite $16$ as $4^{2}$ .

$\log_{8} (\sqrt{4^{2}})$

Step 3.3

Pull terms out from under the radical, assuming positive real numbers.

$\log_{8} (4)$

Step 4

Logarithm base $8$ of $4$ is $\frac{2}{3}$ .

$\frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$