Calculus Examples

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

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

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 2.2

Simplify the exponent.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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Step 2.2.1.1.1.1

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 2.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.1.3.1

Cancel the common factor.

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

Step 2.2.1.1.1.3.2

Rewrite the expression.

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

Step 2.2.1.1.2

Simplify.

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

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

Simplify the expression.

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Step 2.2.2.1.1.1

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

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

Step 2.2.2.1.1.2

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

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

Step 2.2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.2.1

Cancel the common factor.

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

Step 2.2.2.1.2.2

Rewrite the expression.

$x = \pm 2^{3}$

Step 2.2.2.1.3

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

$x = \pm 8$

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.

$x = 8$

Step 2.3.2

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

$x = - 8$

Step 2.3.3

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 $\log_{x} (4) = \frac{2}{3}$ true.

$x = 8$