Precalculus Examples

$\log_{x} (\frac{1}{4}) = - \frac{2}{3}$

Step 1

Rewrite $\log_{x} (\frac{1}{4}) = - \frac{2}{3}$ 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 $b^{y} = x$ .

$x^{- \frac{2}{3}} = \frac{1}{4}$

Step 2

Solve for $x$ .

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Step 2.1

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

$\frac{1}{x^{\frac{2}{3}}} = \frac{1}{4}$

Step 2.2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 \cdot 4 = x^{\frac{2}{3}} \cdot 1$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Rewrite the equation as $x^{\frac{2}{3}} \cdot 1 = 1 \cdot 4$ .

$x^{\frac{2}{3}} \cdot 1 = 1 \cdot 4$

Step 2.3.2

Move the terms containing $x$ to the left side and simplify.

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Step 2.3.2.1

Multiply $x^{\frac{2}{3}}$ by $1$ .

$x^{\frac{2}{3}} = 1 \cdot 4$

Step 2.3.2.2

Multiply $4$ by $1$ .

$x^{\frac{2}{3}} = 4$

Step 2.3.3

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Step 2.3.4

Simplify the exponent.

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Step 2.3.4.1

Simplify the left side.

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Step 2.3.4.1.1

Simplify ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.3.4.1.1.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.3.4.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Step 2.3.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Step 2.3.4.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 4^{\frac{3}{2}}$

Step 2.3.4.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.3.4.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 4^{\frac{3}{2}}$

Step 2.3.4.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm 4^{\frac{3}{2}}$

Step 2.3.4.1.1.2

Simplify.

$x = \pm 4^{\frac{3}{2}}$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $\pm 4^{\frac{3}{2}}$ .

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Step 2.3.4.2.1.1

Simplify the expression.

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Step 2.3.4.2.1.1.1

Rewrite $4$ as $2^{2}$ .

$x = \pm {(2^{2})}^{\frac{3}{2}}$

Step 2.3.4.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm 2^{2 (\frac{3}{2})}$

Step 2.3.4.2.1.2

Cancel the common factor of $2$ .

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Step 2.3.4.2.1.2.1

Cancel the common factor.

$x = \pm 2^{2 (\frac{3}{2})}$

Step 2.3.4.2.1.2.2

Rewrite the expression.

$x = \pm 2^{3}$

Step 2.3.4.2.1.3

Raise $2$ to the power of $3$ .

$x = \pm 8$

Step 2.3.5

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

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Step 2.3.5.1

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

$x = 8$

Step 2.3.5.2

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

$x = - 8$

Step 2.3.5.3

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

$x = 8, - 8$

Step 3

Exclude the solutions that do not make $\log_{x} (\frac{1}{4}) = - \frac{2}{3}$ true.

$x = 8$