Precalculus Examples

$\log_{x} (512) = \frac{3}{2}$

Step 1

Rewrite $\log_{x} (512) = \frac{3}{2}$ 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 $b^{y} = x$ .

$x^{\frac{3}{2}} = 512$

Step 2

Solve for $x$ .

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Step 2.1

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

${(x^{\frac{3}{2}})}^{\frac{2}{3}} = 512^{\frac{2}{3}}$

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

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Step 2.2.1.1.1

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

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Step 2.2.1.1.1.1

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

$x^{\frac{3}{2} \cdot \frac{2}{3}} = 512^{\frac{2}{3}}$

Step 2.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{3}{2} \cdot \frac{2}{3}} = 512^{\frac{2}{3}}$

Step 2.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{2} \cdot 2} = 512^{\frac{2}{3}}$

Step 2.2.1.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 512^{\frac{2}{3}}$

Step 2.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = 512^{\frac{2}{3}}$

Step 2.2.1.1.2

Simplify.

$x = 512^{\frac{2}{3}}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $512^{\frac{2}{3}}$ .

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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 $512$ as $8^{3}$ .

$x = {(8^{3})}^{\frac{2}{3}}$

Step 2.2.2.1.1.2

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

$x = 8^{3 (\frac{2}{3})}$

Step 2.2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.2.1.2.1

Cancel the common factor.

$x = 8^{3 (\frac{2}{3})}$

Step 2.2.2.1.2.2

Rewrite the expression.

$x = 8^{2}$

Step 2.2.2.1.3

Raise $8$ to the power of $2$ .

$x = 64$