Precalculus Examples

$\log_{2} (4 x - 4) - \log_{2} (3 x^{2} + x - 4) + \log_{2} (3 x + 4)$

Step 1

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

$\log_{2} (\frac{4 x - 4}{3 x^{2} + x - 4}) + \log_{2} (3 x + 4)$

Step 2

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

$\log_{2} (\frac{4 x - 4}{3 x^{2} + x - 4} (3 x + 4))$

Step 3

Factor $4$ out of $4 x - 4$ .

Tap for more steps...

Step 3.1

Factor $4$ out of $4 x$ .

$\log_{2} (\frac{4 (x) - 4}{3 x^{2} + x - 4} (3 x + 4))$

Step 3.2

Factor $4$ out of $- 4$ .

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

Step 3.3

Factor $4$ out of $4 (x) + 4 (- 1)$ .

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

Step 4

Factor by grouping.

Tap for more steps...

Step 4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

Tap for more steps...

Step 4.1.1

Multiply by $1$ .

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

Step 4.1.2

Rewrite $1$ as $- 3$ plus $4$

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

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

Step 5

Cancel the common factor of $3 x + 4$ .

Tap for more steps...

Step 5.1

Factor $3 x + 4$ out of $(x - 1) (3 x + 4)$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

$\log_{2} (\frac{4 (x - 1)}{x - 1})$

Step 6

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\log_{2} (\frac{4 (x - 1)}{x - 1})$

Step 6.2

Divide $4$ by $1$ .

$\log_{2} (4)$

Step 7

Logarithm base $2$ of $4$ is $2$ .

$2$