Algebra Examples

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

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 2

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

$\log_{2} (\frac{6 \cdot 2}{8})$

Step 3

Reduce the expression $\frac{6 \cdot 2}{8}$ by cancelling the common factors.

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Factor $2$ out of $6 \cdot 2$ .

$\log_{2} (\frac{2 (3 \cdot 2)}{8})$

Factor $2$ out of $8$ .

$\log_{2} (\frac{2 (3 \cdot 2)}{2 (4)})$

Cancel the common factor.

$\log_{2} (\frac{2 (3 \cdot 2)}{2 \cdot 4})$

Rewrite the expression.

$\log_{2} (\frac{3 \cdot 2}{4})$

Step 4

Reduce the expression $\frac{3 \cdot 2}{4}$ by cancelling the common factors.

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Factor $2$ out of $3 \cdot 2$ .

$\log_{2} (\frac{2 \cdot 3}{4})$

Factor $2$ out of $4$ .

$\log_{2} (\frac{2 \cdot 3}{2 \cdot 2})$

Cancel the common factor.

$\log_{2} (\frac{2 \cdot 3}{2 \cdot 2})$

Rewrite the expression.

$\log_{2} (\frac{3}{2})$

Step 5

Rewrite $\frac{3}{2}$ as $3 \cdot 2^{- 1}$ .

$\log_{2} (3 \cdot 2^{- 1})$

Step 6

Rewrite $\log_{2} (3 \cdot 2^{- 1})$ as $\log_{2} (3) + \log_{2} (2^{- 1})$ .

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

Step 7

Use logarithm rules to move $- 1$ out of the exponent.

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

Step 8

Logarithm base $2$ of $2$ is $1$ .

$\log_{2} (3) - 1 \cdot 1$

Step 9

Multiply $- 1$ by $1$ .

$\log_{2} (3) - 1$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (3) - 1$

Decimal Form:

$0.58496250 \dots$