Solve for x log base 3 of x^2+6x = log base 3 of 2x+12

$\log_{3} (x^{2} + 6 x) = \log_{3} (2 x + 12)$

Step 1

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} + 6 x = 2 x + 12$

Step 2

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

$x^{2} + 6 x - 2 x = 12$

Subtract $2 x$ from $6 x$ .

$x^{2} + 4 x = 12$

Subtract $12$ from both sides of the equation.

$x^{2} + 4 x - 12 = 0$

Factor $x^{2} + 4 x - 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $4$ .

$- 2, 6$

Write the factored form using these integers.

$(x - 2) (x + 6) = 0$

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

$x - 2 = 0$

$x + 6 = 0$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x - 2) (x + 6) = 0$ true.

$x = 2, - 6$

Step 3

Exclude the solutions that do not make $\log_{3} (x^{2} + 6 x) = \log_{3} (2 x + 12)$ true.

$x = 2$

Solve for x log base 3 of x^2+6x = log base 3 of 2x+12

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