Precalculus Examples

$- 2 \log_{4} (x) = \log_{4} (9)$

Step 1

Simplify $- 2 \log_{4} (x)$ by moving $- 2$ inside the logarithm.

$\log_{4} (x^{- 2}) = \log_{4} (9)$

Step 2

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

$x^{- 2} = 9$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{2}} = 9$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2}, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 3.3

Multiply each term in $\frac{1}{x^{2}} = 9$ by $x^{2}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{1}{x^{2}} = 9$ by $x^{2}$ .

$\frac{1}{x^{2}} x^{2} = 9 x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $x^{2}$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1}{x^{2}} x^{2} = 9 x^{2}$

Step 3.3.2.1.2

Rewrite the expression.

$1 = 9 x^{2}$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $9 x^{2} = 1$ .

$9 x^{2} = 1$

Step 3.4.2

Divide each term in $9 x^{2} = 1$ by $9$ and simplify.

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Step 3.4.2.1

Divide each term in $9 x^{2} = 1$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{1}{9}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $9$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{1}{9}$

Step 3.4.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{9}$

Step 3.4.3

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

$x = \pm \sqrt{\frac{1}{9}}$

Step 3.4.4

Simplify $\pm \sqrt{\frac{1}{9}}$ .

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Step 3.4.4.1

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{9}}$

Step 3.4.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{9}}$

Step 3.4.4.3

Simplify the denominator.

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Step 3.4.4.3.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{1}{\sqrt{3^{2}}}$

Step 3.4.4.3.2

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

$x = \pm \frac{1}{3}$

Step 3.4.5

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

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Step 3.4.5.1

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

$x = \frac{1}{3}$

Step 3.4.5.2

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

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

Step 3.4.5.3

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

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

Step 4

Exclude the solutions that do not make $- 2 \log_{4} (x) = \log_{4} (9)$ true.

$x = \frac{1}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}$

Decimal Form:

$x = 0.\overline{3}$