Calculus Examples

$\log_{8} (2) + \log_{4} (2)$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{3} + \log_{4} (2)$

Step 1.2

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\frac{1}{3} + \frac{1}{2}$

Step 2

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2}$

Step 3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}$

Step 4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$\frac{2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}$

Step 4.2

Multiply $3$ by $2$ .

$\frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}$

Step 4.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\frac{2}{6} + \frac{3}{2 \cdot 3}$

Step 4.4

Multiply $2$ by $3$ .

$\frac{2}{6} + \frac{3}{6}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 + 3}{6}$

Step 6

Add $2$ and $3$ .

$\frac{5}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{6}$

Decimal Form:

$0.8\overline{3}$