Solve for x log base 2 of 3x^2+4- log base 2 of 2x+6=3

$\log_{2} (3 x^{2} + 4) - \log_{2} (2 x + 6) = 3$

Simplify the left side.

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

$\log_{2} (\frac{3 x^{2} + 4}{2 x + 6}) = 3$

Factor $2$ out of $2 x + 6$ .

Factor $2$ out of $2 x$ .

$\log_{2} (\frac{3 x^{2} + 4}{2 (x) + 6}) = 3$

Factor $2$ out of $6$ .

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

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

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

Rewrite $\log_{2} (\frac{3 x^{2} + 4}{2 (x + 3)}) = 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

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

Cross multiply to remove the fraction.

$3 x^{2} + 4 = 2^{3} (2 (x + 3))$

Simplify $2^{3} (2 (x + 3))$ .

Multiply $2^{3}$ by $2$ by adding the exponents.

Move $2$ .

$3 x^{2} + 4 = 2 \cdot 2^{3} (x + 3)$

Multiply $2$ by $2^{3}$ .

Raise $2$ to the power of $1$ .

$3 x^{2} + 4 = 2^{1} \cdot 2^{3} (x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{2} + 4 = 2^{1 + 3} (x + 3)$

Add $1$ and $3$ .

$3 x^{2} + 4 = 2^{4} (x + 3)$

Raise $2$ to the power of $4$ .

$3 x^{2} + 4 = 16 (x + 3)$

Apply the distributive property.

$3 x^{2} + 4 = 16 x + 16 \cdot 3$

Multiply $16$ by $3$ .

$3 x^{2} + 4 = 16 x + 48$

Subtract $16 x$ from both sides of the equation.

$3 x^{2} + 4 - 16 x = 48$

Move all terms not containing $x$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$3 x^{2} - 16 x = 48 - 4$

Subtract $4$ from $48$ .

$3 x^{2} - 16 x = 44$

Subtract $44$ from both sides of the equation.

$3 x^{2} - 16 x - 44 = 0$

Factor by grouping.

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 $a \cdot c = 3 \cdot - 44 = - 132$ and whose sum is $b = - 16$ .

Factor $- 16$ out of $- 16 x$ .

$3 x^{2} - 16 x - 44 = 0$

Rewrite $- 16$ as $6$ plus $- 22$

$3 x^{2} + (6 - 22) x - 44 = 0$

Apply the distributive property.

$3 x^{2} + 6 x - 22 x - 44 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} + 6 x) - 22 x - 44 = 0$

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

$3 x (x + 2) - 22 (x + 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$(x + 2) (3 x - 22) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$3 x - 22 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $3 x - 22$ equal to $0$ and solve for $x$ .

Set $3 x - 22$ equal to $0$ .

$3 x - 22 = 0$

Solve $3 x - 22 = 0$ for $x$ .

Add $22$ to both sides of the equation.

$3 x = 22$

Divide each term in $3 x = 22$ by $3$ and simplify.

Divide each term in $3 x = 22$ by $3$ .

$\frac{3 x}{3} = \frac{22}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{22}{3}$

Divide $x$ by $1$ .

$x = \frac{22}{3}$

The final solution is all the values that make $(x + 2) (3 x - 22) = 0$ true.

$x = - 2, \frac{22}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = - 2, \frac{22}{3}$

Decimal Form:

$x = - 2, 7.\overline{3}$

Mixed Number Form:

$x = - 2, 7 \frac{1}{3}$

Solve for x log base 2 of 3x^2+4- log base 2 of 2x+6=3

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