Solve for x log base 6 of 3x^2-7- log base 6 of x+7=0

$\log_{6} (3 x^{2} - 7) - \log_{6} (x + 7) = 0$

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

$\log_{6} (\frac{3 x^{2} - 7}{x + 7}) = 0$

Rewrite $\log_{6} (\frac{3 x^{2} - 7}{x + 7}) = 0$ 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 $b^{y} = x$ .

$6^{0} = \frac{3 x^{2} - 7}{x + 7}$

Cross multiply to remove the fraction.

$3 x^{2} - 7 = 6^{0} (x + 7)$

Simplify $6^{0} (x + 7)$ .

Anything raised to $0$ is $1$ .

$3 x^{2} - 7 = 1 (x + 7)$

Multiply $x + 7$ by $1$ .

$3 x^{2} - 7 = x + 7$

Subtract $x$ from both sides of the equation.

$3 x^{2} - 7 - x = 7$

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

Add $7$ to both sides of the equation.

$3 x^{2} - x = 7 + 7$

Add $7$ and $7$ .

$3 x^{2} - x = 14$

Subtract $14$ from both sides of the equation.

$3 x^{2} - x - 14 = 0$

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 $a \cdot c = 3 \cdot - 14 = - 42$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$3 x^{2} - (x) - 14 = 0$

Rewrite $- 1$ as $6$ plus $- 7$

$3 x^{2} + (6 - 7) x - 14 = 0$

Apply the distributive property.

$3 x^{2} + 6 x - 7 x - 14 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} + 6 x) - 7 x - 14 = 0$

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

$3 x (x + 2) - 7 (x + 2) = 0$

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

$(x + 2) (3 x - 7) = 0$

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

$x + 2 = 0$

$3 x - 7 = 0$

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 $3 x - 7$ equal to $0$ and solve for $x$ .

Set $3 x - 7$ equal to $0$ .

$3 x - 7 = 0$

Solve $3 x - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$3 x = 7$

Divide each term in $3 x = 7$ by $3$ and simplify.

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Divide $x$ by $1$ .

$x = \frac{7}{3}$

The final solution is all the values that make $(x + 2) (3 x - 7) = 0$ true.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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

Mixed Number Form:

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

Solve for x log base 6 of 3x^2-7- log base 6 of x+7=0

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