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

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

Write in exponential form.

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 3$ , $x = x^{6}$ , and $y = 12$ .

$b = 3$

$x = x^{6}$

$y = 12$

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$3^{12} = x^{6}$

Solve for $x$ .

Rewrite the equation as $x^{6} = 3^{12}$ .

$x^{6} = 3^{12}$

Take the 6th root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[6]{3^{12}}$

Simplify $\pm \sqrt[6]{3^{12}}$ .

Raise $3$ to the power of $12$ .

$x = \pm \sqrt[6]{531441}$

Rewrite $531441$ as $9^{6}$ .

$x = \pm \sqrt[6]{9^{6}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 9$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 9$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 9$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 9, - 9$

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

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