Solve for x log base 2 of x-6+ log base 2 of 4x+7=3

$\log_{2} (x - 6) + \log_{2} (4 x + 7) = 3$

Simplify the left side.

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

$\log_{2} ((x - 6) (4 x + 7)) = 3$

Expand $(x - 6) (4 x + 7)$ using the FOIL Method.

Apply the distributive property.

$\log_{2} (x (4 x + 7) - 6 (4 x + 7)) = 3$

Apply the distributive property.

$\log_{2} (x (4 x) + x \cdot 7 - 6 (4 x + 7)) = 3$

Apply the distributive property.

$\log_{2} (x (4 x) + x \cdot 7 - 6 (4 x) - 6 \cdot 7) = 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\log_{2} (4 x \cdot x + x \cdot 7 - 6 (4 x) - 6 \cdot 7) = 3$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\log_{2} (4 (x \cdot x) + x \cdot 7 - 6 (4 x) - 6 \cdot 7) = 3$

Multiply $x$ by $x$ .

$\log_{2} (4 x^{2} + x \cdot 7 - 6 (4 x) - 6 \cdot 7) = 3$

Move $7$ to the left of $x$ .

$\log_{2} (4 x^{2} + 7 \cdot x - 6 (4 x) - 6 \cdot 7) = 3$

Multiply $4$ by $- 6$ .

$\log_{2} (4 x^{2} + 7 x - 24 x - 6 \cdot 7) = 3$

Multiply $- 6$ by $7$ .

$\log_{2} (4 x^{2} + 7 x - 24 x - 42) = 3$

Subtract $24 x$ from $7 x$ .

$\log_{2} (4 x^{2} - 17 x - 42) = 3$

Rewrite $\log_{2} (4 x^{2} - 17 x - 42) = 3$ 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 $b^{y} = x$ .

$2^{3} = 4 x^{2} - 17 x - 42$

Solve for $x$ .

Rewrite the equation as $4 x^{2} - 17 x - 42 = 2^{3}$ .

$4 x^{2} - 17 x - 42 = 2^{3}$

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

$4 x^{2} - 17 x - 42 = 8$

Subtract $8$ from both sides of the equation.

$4 x^{2} - 17 x - 42 - 8 = 0$

Subtract $8$ from $- 42$ .

$4 x^{2} - 17 x - 50 = 0$

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 $a \cdot c = 4 \cdot - 50 = - 200$ and whose sum is $b = - 17$ .

Factor $- 17$ out of $- 17 x$ .

$4 x^{2} - 17 x - 50 = 0$

Rewrite $- 17$ as $8$ plus $- 25$

$4 x^{2} + (8 - 25) x - 50 = 0$

Apply the distributive property.

$4 x^{2} + 8 x - 25 x - 50 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} + 8 x) - 25 x - 50 = 0$

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

$4 x (x + 2) - 25 (x + 2) = 0$

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

$(x + 2) (4 x - 25) = 0$

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

$x + 2 = 0$

$4 x - 25 = 0$

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 $4 x - 25$ equal to $0$ and solve for $x$ .

Set $4 x - 25$ equal to $0$ .

$4 x - 25 = 0$

Solve $4 x - 25 = 0$ for $x$ .

Add $25$ to both sides of the equation.

$4 x = 25$

Divide each term in $4 x = 25$ by $4$ and simplify.

Divide each term in $4 x = 25$ by $4$ .

$\frac{4 x}{4} = \frac{25}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{25}{4}$

Divide $x$ by $1$ .

$x = \frac{25}{4}$

The final solution is all the values that make $(x + 2) (4 x - 25) = 0$ true.

$x = - 2, \frac{25}{4}$

Exclude the solutions that do not make $\log_{2} (x - 6) + \log_{2} (4 x + 7) = 3$ true.

$x = \frac{25}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{25}{4}$

Decimal Form:

$x = 6.25$

Mixed Number Form:

$x = 6 \frac{1}{4}$

Solve for x log base 2 of x-6+ log base 2 of 4x+7=3

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