Algebra Examples

$\log_{3 x} (324) = 2$

Step 1

Rewrite $\log_{3 x} (324) = 2$ 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 $b^{y} = x$ .

${(3 x)}^{2} = 324$

Step 2

Solve for $x$ .

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Step 2.1

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

$3 x = \pm \sqrt{324}$

Step 2.2

Simplify $\pm \sqrt{324}$ .

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Step 2.2.1

Rewrite $324$ as $18^{2}$ .

$3 x = \pm \sqrt{18^{2}}$

Step 2.2.2

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

$3 x = \pm 18$

Step 2.3

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

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Step 2.3.1

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

$3 x = 18$

Step 2.3.2

Divide each term in $3 x = 18$ by $3$ and simplify.

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Step 2.3.2.1

Divide each term in $3 x = 18$ by $3$ .

$\frac{3 x}{3} = \frac{18}{3}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{18}{3}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{3}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $18$ by $3$ .

$x = 6$

Step 2.3.3

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

$3 x = - 18$

Step 2.3.4

Divide each term in $3 x = - 18$ by $3$ and simplify.

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Step 2.3.4.1

Divide each term in $3 x = - 18$ by $3$ .

$\frac{3 x}{3} = \frac{- 18}{3}$

Step 2.3.4.2

Simplify the left side.

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Step 2.3.4.2.1

Cancel the common factor of $3$ .

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Step 2.3.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 18}{3}$

Step 2.3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 18}{3}$

Step 2.3.4.3

Simplify the right side.

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Step 2.3.4.3.1

Divide $- 18$ by $3$ .

$x = - 6$

Step 2.3.5

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

$x = 6, - 6$

Step 3

Exclude the solutions that do not make $\log_{3 x} (324) = 2$ true.

$x = 6$