Algebra Examples

$\log_{2} (56) + 2 \log_{2} (12) - \log_{2} (63)$

Step 1

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

$2 \log_{2} (12) + \log_{2} (\frac{56}{63})$

Step 2

Simplify each term.

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Step 2.1

Simplify $2 \log_{2} (12)$ by moving $2$ inside the logarithm.

$\log_{2} (12^{2}) + \log_{2} (\frac{56}{63})$

Step 2.2

Raise $12$ to the power of $2$ .

$\log_{2} (144) + \log_{2} (\frac{56}{63})$

Step 2.3

Cancel the common factor of $56$ and $63$ .

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Step 2.3.1

Factor $7$ out of $56$ .

$\log_{2} (144) + \log_{2} (\frac{7 (8)}{63})$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $7$ out of $63$ .

$\log_{2} (144) + \log_{2} (\frac{7 \cdot 8}{7 \cdot 9})$

Step 2.3.2.2

Cancel the common factor.

$\log_{2} (144) + \log_{2} (\frac{7 \cdot 8}{7 \cdot 9})$

Step 2.3.2.3

Rewrite the expression.

$\log_{2} (144) + \log_{2} (\frac{8}{9})$

Step 3

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

$\log_{2} (144 (\frac{8}{9}))$

Step 4

Cancel the common factor of $9$ .

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Step 4.1

Factor $9$ out of $144$ .

$\log_{2} (9 (16) \frac{8}{9})$

Step 4.2

Cancel the common factor.

$\log_{2} (9 \cdot 16 \frac{8}{9})$

Step 4.3

Rewrite the expression.

$\log_{2} (16 \cdot 8)$

Step 5

Simplify the expression.

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Step 5.1

Multiply $16$ by $8$ .

$\log_{2} (128)$

Step 5.2

Logarithm base $2$ of $128$ is $7$ .

$7$