Simplify/Condense log base 2 of 27+ log base 2 of 4-2* log base 2 of 3

$\log_{2} (27) + \log_{2} (4) - 2 \cdot \log_{2} (3)$

Step 1

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

$\log_{2} (27 \cdot 4) - 2 \cdot \log_{2} (3)$

Step 2

Simplify each term.

Multiply $27$ by $4$ .

$\log_{2} (108) - 2 \cdot \log_{2} (3)$

Simplify $- 2 \log_{2} (3)$ by moving $2$ inside the logarithm.

$\log_{2} (108) - \log_{2} (3^{2})$

Raise $3$ to the power of $2$ .

$\log_{2} (108) - \log_{2} (9)$

Step 3

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

$\log_{2} (\frac{108}{9})$

Step 4

Divide $108$ by $9$ .

$\log_{2} (12)$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (12)$

Decimal Form:

$3.58496250 \dots$